DYNAMIC AND ADAPTIVE TESSELLATION OF BÉZIER SURFACES

Transcription

1 DYNAMIC AND ADAPTIVE TESSELLATION OF BÉZIER SURFACES R. Concheiro, M. Amor Univeriy of A Coruña, Spain rconcheiro@udc.e, margamor@udc.e M. Bóo Univeriy of Saniago de Compoela, Spain monerra.boo@uc.e Keyword: Abrac: M. Dogge Lund Univeriy, Sweden mike@c.lh.e Adapive eellaion, GPU, Bézier urface. Bézier urface offer powerful mechanim o decribe high qualiy model in compuer graphic. In hi paper we preen a mehodology for he implemenaion of he adapive eellaion of Bézier model on he GPU. Teellaion i performed wih variable reoluion inide he pach o permi he ue of mehe wih a lower number of riangle bu preerving a high viualizaion qualiy. Primiive are dynamically generaed according o he reul of local ubdiviion e. The localiy of he deciion aure ha coniguou riangle are coherenly ubdivided.the reuling procedure i efficien, imple and generae he eellaion paern of each Bézier urface dynamically. A conequence eellaion of complex model can be performed in real ime. 1 INTRODUCTION Bézier repreenaion have been widely employed a a andard way of deigning complex cene wih very good qualiy reul. In many applicaion involving CAD/CAM, virual realiy, animaion and viualizaion, objec model are decribed in erm of Bézier urface. The excellen mahemaical and algorihmic properie (Piegl and Tiller, 1997), combined wih ucceful indurial applicaion, have conribued o he populariy of hi repreenaion. Tradiionally, for he rendering proce he Bézier model are eellaed on he CPU (Cenral Proceing Uni) and he e of generaed riangle i en o he GPU (Graphic Proceing Uni). The CPU-GPU bu can become a boleneck in hi approach due o he large number of riangle generaed for high qualiy model. Nowaday, here are ome approach o perform he eellaion of he parameric model direcly on he GPU. In hee propoal he eellaion level i eleced per pach (Guhe e al., 2005; Concheiro e al., 2010) or per e of pache (Dyken e al., 2009). Anoher eellaion approach i preened in (Eienacher e al., 2009; Schwarz and Samminger, 2009) where he eellaion i performed following a GPGPU raegy (General-Purpoe Compuaion on GPU). In a imilar way, all hee propoal are adapive a he pach level, ha i, once he reoluion for a pach i deermined, he pach i ubdivided in a uniform way (wih pecific modificaion in he border of he pach o aure no hole or crack beween neighbor pache). Unil recenly, he programmable verex and pixel hader found in GPU could only operae on exiing daa. The cene changed a few year ago wih he inroducion of a new programmable uni, he geomery hader and he inroducion of new eellaion age (hull hader, eellaor and domain hader) (Ni and Caaño, 2009). Some adapive eellaion propoal for riangle mehe exploiing geomery hader have been developed (Bóo e al., 2011; Lorenz and Döllner, 2008) and no one, a far a we know, i focued on he adapive eellaion of parameric model a he riangle level. The cene did no change wih he inroducion of he new eellaor uni becaue i applied a fixed and emi-regular paern alo a he pach level. In hi paper we preen a new mehod o adapively eellae Bézier urface on he GPU. Our Dynamic and Adapive Bézier Teellaion (DABT) propoal permi he minimizaion of he number of riangle o be proceed wihou reducing he qualiy of he final image. To increae he adapive proper-

2 Iniial riangle Teellaion Paern Sampling poin L=3 Te Uni Inered verice Meh Uni Figure 1: Scheme of he DABT algorihm Final riangle ie of previou propoal, neighbor riangle can be eellaed wih a differen reoluion wihou crack. Therefore, our mehod eellae Bézier urface in a adapive way, o real-ime rendering of complex model can be achieved. 2 DYNAMIC AND ADAPTIVE BÉZIER TESSELLATION In hi ecion we preen our Dynamic and Adapive Bézier Teellaion (DABT) propoal. Our raegy perform he adapive eellaion of he Bézier urface by compuing he eellaion paern on-hefly wihou employing a e of pre-compued paern. The objecive i a freely adapive eellaion inide each pach where he reoluion and number of riangle generaed can be eleced a a rade off beween qualiy and compuaional requiremen. The mehodology we propoe i baed on local e, he reuling model ha no crack or hole, neiher inide each pach nor beween neighbor pache. Specifically, our DABT mehod i baed on hree differen key propoal: he uilizaion of a fixed eellaion paern ha guide he adapive eellaion, he applicaion of local e and an efficien mehing procedure o reconruc he reuling meh once he e are applied. Figure 1 chemaically how hee hree key core of he algorihm. 2.1 Uilizaion of a Fixed Paern o Guide he Adapive Teellaion Procedure The objecive of he propoal i exploiing he large number of core available in curren GPU. Wih hi objecive in mind he pache of he model are iniially eellaed and he coare riangle employed a inpu primiive for he applicaion. Teellaion algorihm wih a recurive naure have differen diadvanage. For hi reaon he DABT mehod employ a non recurive raegy baed on he uilizaion of a non adapive eellaion paern a a bai for he adapive cae. Once he eellaion level of he paern i eleced only he V2 V1 E1,2 E1,3 E2,3 V3 Figure 2: Example of riangle wih hree differen reoluion area poiion in hi eellaion paern are evaluaed for condiional inerion a each poiion. Figure 1 how he eellaion paern we employ for a level of reoluion L = 3. The original coare riangle i depiced wih bold line and he ampling poin correponding o he candidae verice wih a cro. The parameric coordinae (u B,v B ) of he ampling poin aociaed wih a candidae verex V B ha lie on he Bézier urface are compued hrough i barycenric coordinae wih: V B (w i,w j,w k ) = w i (u 1,v 1 ) + w j (u 2,v 2 )+ w k (u 3,v 3 ) where (u 1,v 1 ), (u 2,v 2 ) and (u 3,v 3 ) are he parameric coordinae of he verice of he iniial coare riangle and (w i,w j,w k ) are he barycenric coordinae of he candidae verex. The barycenric value are in he inerval [0,1] and verify w i = i δw, w j = j δw and w k = k δw wih i, j,k = 0,...L + 1 and δw = L+1 1.The eellaion i performed in he parameric domain, for he pecific cae of a bi-cubic Bézier urface he following equaion i evaluaed: Q(u,v) = [u 3 u 2 u 1] B 0,0 B 0,1 B 0,2 B 0,3 B 1,0 B 1,1 B 1,2 B 1,3 B 2,0 B 2,1 B 2,2 B 2,3 B 3,0 B 3,1 B 3,2 B 3, In order o enrich he adapive eellaion poibiliie, we have developed a mehod for he aignaion of differen reoluion level o neighbor riangle. Wih hi non uniform approach, he reoluion level can be dynamically eleced and modified along he pach. Specifically a reoluion level i eleced per riangle edge o one riangle could have hree reoluion level. In order o apply hree level per riangle, each one would be applied o one hird of he riangle. Figure 2 how a riangle where hree differen reoluion level coexi. In hi example he area E 1,2 follow a eellaion paern L = 3 ha mean v 3 v 2 v 1

3 ha hree row of candidae verice have o be analyzed. Area E 1,3 i ubdivided according o a eellaion paern L = 2, ha i, wo row of candidae verice have o be conidered. Finally, E 2,3 i eellaed wih a reoluion level L = Selecion of Te Employed o Guide he Adapive Teellaion The candidae verice are condiionally inered according o he reul of pecific eellaion e (ee Figure 1). We have evaluaed differen e o guide he eellaion procedure. Thee e meaure, hrough he uilizaion of differen qualiy hrehold, he incremen of qualiy of he meh if a given verex i inered Diance e Thi e analyze he diance beween he riangle meh and he Bézier urface. Specifically he diance beween a ampling poin on he riangle meh and he correponding poin on he Bézier urface i analyzed. If he diance i mall enough, he riangle meh i conidered a good approximaion of he urface o no verex i inered. On he conrary, if he diance i large he verex i inroduced a hi will increae he qualiy of he final image. The e i given by: diance = [ V S V B > diance ] where diance i a qualiy hrehold ha i eleced in funcion of he qualiy/iming requiremen of he applicaion, V S i he correponding ampling poin on he coare riangle by inerpolaing he poiion of he original verice and V B i he coordinae of he candidae verex on he Bézier urface obained by uing Equaion Vecor deviaion fla e The objecive of hi e i employing he curvaure of he urface a a parameer o guide he eellaion procedure. Wih hi e candidae verice on fla area are no conidered for inerion a he qualiy of he urface would no be incremened. To check he curvaure of he urface a imple vecor deviaion fla e (Epino e al., 2007) can be employed. In order o reduce he coly compuaion of all candidae verice our flane e follow a per edge philoophy. In our propoal each candidae poiion aociaed wih he coare edge i analyzed and he deciion performed i applied o all candidae poiion of he ame row and reoluion area.then, wih hi echnique he curvaure of he urface i eimaed only for he ampling poin on he edge of he coare riangle and he reul are employed in he inerior of he projecion. Specifically he e coni on he following ep: 1. Calculaion and normalizaion of vecor A = V 1 V 2, B = V B V 1 and C = V B V 2, where V 1 and V 2 are he exreme poin of each edge. 2. Compuaion of he unigned do produc BA and CA 3. Comparion beween he do produc and a hrehold, f la. If one of hem i maller han he hrehold, he new verex i inered: f la = ( BA < f la ) OR ( CA < f la ) Lengh e In geomeric deign applicaion raher han uing a very high degree urface o approximae a very complex urface, i i more common o break he urface up ino everal imple urface. The uual way o make larger and complex urface i o connec up a e of low degree Bézier urface. The e we have employed exploi hi ypical low degree/curvaure of he Bézier urface. Specifically, he e i baed on he uilizaion of he lengh of he coare riangle edge a a meaure of he curvaure of he Bézier urface in he correponding area. Due o low degree of each Bézier urface and a he verice of he coare riangle meh lie on he urface, if he coare riangle i mall hen i can be conidered a good approximaion o he urface. Specifically, if he wo verice of one edge are cloe enough, he incluion of addiional verice on ha edge will no increae he qualiy of he final meh. Thi e work on he edge bai a i i only applied for he ampling poin on he edge of he coare riangle. In cae a verex correponding o he he edge i inered, he verice on he ame row are direcly inered. Noe ha he e i baed on he analyi of he original verice of he riangle and doe no require he compuaion of he candidae verice. A conequence, he compuaional requiremen of hi new e are very low. Specifically, o e if a candidae verex V B ha o be inered in he edge wih verice V 1 and V 2 he following analyi i performed: lengh = ( V 1 V S > lengh ) AND ( V 2 V S > lengh ) ha mean ha he poin V B i inered only when he diance of he correponding ampling poin on he riangle V S o boh exreme verice V 1 and V 2 i larger han a hrehold lengh.

4 2.3 Teellaion procedure In hi ecion, we decribe he eellaion procedure employed o generae he final riangle once he inered verice have been deermined (ee Figure 1). A imilar raegy wa employed in (Bóo e al., 2011) for he adapive eellaion of generic riangle mehe. Thi eellaion procedure i baed on he claificaion of he inered verice in rip and he efficien managemen of he reuling li of verice. The objecive in mind i generaing he riangle rucure in a direc way from he irregular paern obained wih he evaluaion of he ubdiviion e. A he non adapive eellaion paern are characerized by a row rucure our algorihm inheri hi characeriic and claifie he reuling inered verice in row. A uple of L + 2 li Sv = (Sv 1,,Sv L+2 ) correponding o he L + 2 rip of verice are ued. Each li include he verice inered in each rip and poiion of non-inered are empy. An example of adapive eellaion i depiced in Figure 3(a) where he candidae poiion are labeled wih number and he verice finally inered are indicaed wih do. In hi cae, he final eellaion paern can be repreened wih he following li: Sv 1 = {1}, Sv 2 = {}, Sv 3 = {4,5,6}, Sv 4 = {9,10}, Sv 5 = {11,13,15} The objecive i generaing he final riangle by connecing he verice in wo conecuive li. However he direc applicaion of hi echnique would lead o he generaion of no deired overlapping riangle. To avoid hi problem, wo modificaion are inroduced o he repreenaion: reue of limi verice and incorporaion of new exreme verice. A limi verex i a verex locaed in he original edge of he riangle. If uch a verex doe no exi, an exreme verex i he fir/la verex in he li. Wih repec o he modificaion relaed wih he limi verice, if here i no limi verex in a row a limi verex from a previou row ha o be included. A an example for he li of verice correponding o Figure 3(a) hi modificaion implie ha he Sv 4 li ha o be exended a Sv 4 ={ˆ4, 9, 10} where he reued limi verex i indicaed wih a ha. The reue of limi verice permi he generaion of riangle connecing verice in non conecuive row. A a conequence and in order o avoid an overlap of hee large riangle wih oher local rucure, he incorporaion of new exreme verice are required. A deailed explanaion can be found in (Bóo e al., 2011). The eellaion procedure work by proceing pair of conecuive rip of verice. In hi mehod he riangle are generaed by joining he verice beween conecuive rip (paren-children relaion) (a) Figure 3: Example of meh reconrucion (a) Verice inered (b) Teellaion generaed aking ino accoun he following rule: (b) Two conecuive verice in he ame rip are alway conneced (ibling relaion). Two idenical verice are no conidered for connecion. A reued opening/cloing limi verex ha a limied connecion wih he following non empy rip. Specifically, i can only be conneced wih a non-copied opening/cloing limi verex. Each non reued verex of each rip, conidered a paren, i conneced wih conecuive children in he following rip. The following paren i conneced wih anoher group of conecuive children. There i an overlap of one common child beween wo conecuive paren. The applicaion of hee rule o each pair of exended verex li generae he final eellaion. 3 IMPLEMENTATION OF THE DABT ALGORITHM ON THE GEOMETRY SHADER Our implemenaion procee bi-cubic Bézier urface and exploi he capabiliie of he geomery hader ha permi a fully adapive eellaion. However, he main drawback of he geomery hader i he limiaion of he number of oupu primiive per invocaion, a currenly only bi value can be oupu. The pache of he model are iniially eellaed and he reuling coare riangle are employed a inpu primiive for he applicaion. To do hi, he parameric domain (u,v) i pariioned in N u N v cell of ize 1 N u 1 N v where wo adjoining riangle are generaed per cell. Once he coare meh i exraced, he adapive eellaion algorihm i applied o each coare riangle. A wa previouly indicaed, he eellaion

5 (a) (b) (c) 13 Figure 4: Unified reoluion Luni f ied =11 for a yem wih L=0,1,2,3 procedure i baed on he uilizaion of a non adapive paern o guide he eellaion. Our implemenaion i baed on he row rucure of he eellaion paern o ome modificaion had o be performed o permi he proceing of hree reoluion area per riangle. Specifically and in order o follow a row compuaion raegy, an unified reoluion i eleced and employed. For a yem wih reoluion level L= (0,, Lmax ), he unified reoluion level correpond wih he lea common muliple of (0,, Lmax + 1) minu 1. A an example le u conider a yem wih reoluion L=(0, 1, 2, 3), in hi cae he unified reoluion level i Luni f ied = 11. The barycenric weigh employed for he candidae for verice each reoluion are:w1 = w2 = 13, 32 w3 = 14, 21, 34 being wl he weigh employed for level L. The unified yem of weigh i: w =,,,,,,,,,, where w10 = w31 = w 5, w0 = w3, w1 = w7, w0 = w2 3 and w2 = w8. A reul, a unified yem of weigh and row can be employed for any reoluion level. Figure 4 how he eellaion paern for Luni f ied = 11. In fac, he e uni procee only he poin aociaed wih he reoluion level eleced. In he figure he poin correponding o L=1 (line 7 and 13), L=2 (line 5, 9 and 13) and L=3 (line 4, 7, 10 and 13) are indicaed wih circle. 4 RESULTS In hi ecion we preen he reul of he evaluaion of our propoal in erm of he qualiy of he final image and frame per econd (fp). We have evaluaed our propoal on an Inel Core GHz wih 2 GB of RAM and an ATI Radeon The model employed in he e preened in hi ecion are hown in Figure 5: Teacup in Figure 5 (a), Teapo in Figure 5 (b) and Elephan in Figure 5 (c). Figure 5: Model employed: (a) Teacup, (b) Teapo and (c) Elephan Di a n c ete Fl a e L e n g h e Figure 6: Zoom of he eellaed eacup model obained wih he hree e: Diance, Fla and Lengh The cene we have employed for our e conain replicaed and caled verion of hee model. Table 1 ummarize he reul obained in erm of number of primiive. For he e ummarized in hi able, Lmax = 3 and hree qualiy e of hrehold were employed: High, Medium and Low. To obain hi daa and for each qualiy level, he hrehold for he hree e were eleced o enure a imilar number of riangle for he eellaed model. The econd column include he number of B ezier urface per cene and he number of model replicaed. The hird, fourh and fifh column include he average number Scene\Te Crazy eacup Crazy eapo Elephan herd # urface 260 (10) 960 (30) 8110 (10) Diance e k k k High Qualiy Fla e k k k Lengh e k k k Scene\Te Crazy eacup Crazy eapo Elephan herd # urface 260 (10) 960 (30) 8110 (10) Diance e k k k Medium Qualiy Fla e k k k Lengh e k k k Scene\Te Crazy eacup Crazy eapo Elephan herd # urface 260 (10) 960 (30) 8110 (10) Diance e 15.1 k k k Low Qualiy Fla e k 6.40 k k Lengh e k k k Table 1: Number of riangle generaed wih he differen e preened and Lmax = 3

6 Diance e Lengh e Fla e FPS High Medium Low Figure 7: Error obained wih he eacup model for L max = 3 and hree differen qualiy level Number of original riangle (K) Figure 8: FPS wih eellaion level L max = 3 for a high qualiy hrehold of generaed riangle wih each e preened in Secion 2.2. Specifically he e baed on he diance beween meh and urface i labeled a Diance e, he Vecor deviaion fla e a Fla e while he e baed on he lengh of he riangle coare edge a Lengh e. The e we have performed indicae ha he eellaion mehod produce high qualiy mehe wih no viual arifac. Addiionally he mulireoluion mehod employed and he uilizaion of muliple reoluion area per riangle have produced he expeced good reul. In Figure 6 a zoom how he differen eellaion obained for he eacup model for he hree e: Diance, Fla and Lengh. To furher evaluae and compare he reul in erm of qualiy he numerical error of he reuling mehe wa alo analyzed. We have compued he meh error a an eimaion of he diance beween he real urface (approximaed wih a high reoluion non adapive eellaion of he meh) and he reuling meh obained hrough he adapive echnique. Figure 7 how he meh error for a reuling cene (eacup), wih a reoluion level L max = 3 for hree differen qualiy level. A i can be oberved and a expeced he be reul are obained wih he Diance e. Thi i due o he fac ha i he unique e performed per candidae poiion inead of per edge and, a reul, higher qualiy i obained. Very cloe reul are obained wih he Lengh e while wore reul are obained wih he Fla e. Boh e are performed per edge and he reul indicae ha for low degree urface he Lengh e give a good eimaion of he urface curvaure. The main objecive of he propoal i he realime rendering of complex model wihou reducing he qualiy of he final image. Wih he objecive of eing he iming requiremen of he applicaion we have analyzed a walk-hrough animaion wih he ame movemen of he camera for all e. The final image have a creen reoluion of pixel. Figure 8 how he reul in fp uing a reoluion level L max = 3 for a high qualiy hrehold. The reul indicae very good performance in erm of fp, allowing real-ime adapive eellaion, even for a high number of riangle. For example, for he Lengh Te K riangle were rendered a fp. In hi card and for a large number of riangle he Diance e ha imilar iming requiremen han he oher propoal. Thi i due due o he exploiaion of VLIW wih he uilizaion of hor vecor daa ype (like f loa4) and vecor compuaion. 5 CONCLUSIONS In hi paper we preen a new mehod, Dynamic and Adapive Bézier Teellaion (DABT), for he real ime adapive eellaion of Bézier urface on he GPU. The mehod i baed on he generaion of a iniial coare riangle meh ha approximae he Bézier urface and he adapive eellaion of each reuling riangle in he GPU. The mehodology employed permi applying muliple reoluion o he ame Bézier urface. Thi mean ha neighbor riangle can be proceed wih differen reoluion and no viual arifac are aured. The propoal i baed on hree main raegie: he uilizaion of a fixed eellaion paern o guide he procedure, he uilizaion of local e for he adapive eellaion deciion and an efficien mehing procedure o reconruc he reuling mehe. Wih repec o he e employed, we have included in hi work hree e ha analyze differen urface feaure o guide he eellaion. Specifically we include one e udying he diance meh-urface, oher baed on he urface curvaure analyi and a hird one baed on ize evaluaion. For he algorihm eing purpoe and o evaluae he capabiliie of curren GPU we have implemened our DABT algorihm by exploiing he geomery hader uni. The good reul obained in erm of qualiy and frame per econd, make our propoal an inereing candidae for i real hardware imple-

7 menaion on fuure GPU. ACKNOWLEDGEMENTS Thi work wa uppored by he Xuna de Galicia under projec INCITE08PXIB105161PR and 08TIC0006PR, he Miniry of Science and Innovaion, cofunded by he FEDER fund of he European Union under he gran TIN , and he Conolidaion of Compeiive Reearch Group ref. 2010/06. REFERENCES Bóo, M., Amor, M., Concheiro, R., and Dogge, M. (2011). Dynamic and Adapive Meh Refinemen on he GPU. Inernal Repor. Concheiro, R., Amor, M., and Bóo, M. (2010). Synhei of bézier urface. In GRAPP 10: Inernaional Conference on Compuer Graphic Theory and Applicaion, page Dyken, C., M., R., and Seland, J. (2009). Semi-uniform Adapive Pach Teellaion. Compuer Graphic Forum, 28(8): Eienacher, C., Meyer, Q., and Loop, C. (2009). Real-ime View-dependen Rendering of Parameric Surface. In Proceeding of he 2009 Sympoium on Ineracive 3D Graphic and Game, page Epino, F. J., Bóo, M., Amor, M., and Bruguera, J. D. (2007). Hardware Suppor for Adapive Teellaion of Bézier Surface Baed on Local Te. Journal of Syem Archiecure, 53(4): Guhe, M., Baláz, A., and Klein, R. (2005). GPU-Baed Trimming and Teellaion of NURBS and T-Spline Surface. ACM Tran. Graph., 24(3): Lorenz, H. and Döllner, J. (2008). Dynamic Meh Refinemen on GPU uing Geomery Shader. In Proceeding of he 16-h Inernaional Conference in Cenral Europe on Compuer Graphic, Viualizaion and Compuer Viion Ni, T. and Caaño, I. (2009). Efficien Subiue for Subdiviion Surface. Exhibiion Tech. SIGGRAPH 09 Coure Noe, Piegl, L. and Tiller, W. (1997). The NURBS Book. Springer. Schwarz, M. and Samminger, M. (2009). Fa GPU-baed Adapive Teellaion wih CUDA. Compuer Graphic Forum, 28(2):