Non-niform Sbdivision for B-slines of Arbirary Degree S. Schaefer, R. Goldman We resen an efficien algorihm for sbdividing non-niform B-slines of arbirary degree in a manner similar o he Lane-Riesenfeld sbdivision algorihm for niform B- slines of arbirary degree. Or algorihm consiss of dobling he conrol oins followed by d ronds of non-niform averaging similar o he d ronds of niform averaging in he Lane-Riesenfeld algorihm for niform B-slines of degree d. However, nlike he Lane- Riesenfeld algorihm which follows mos direcly from he coninos convolion formla for he niform B-sline basis fncions, or algorihm follows narally from blossoming. We show ha or kno inserion mehod is simler and more efficien han revios kno inserion algorihms for non-niform B-slines.. Inrodcion Sbdivision crves and srfaces are biqios in Comer Grahics and Geomeric Modeling. Sbdivision recrsively refines simle olygonal shaes which, if he sbdivision rles are chosen correcly, converge in he limi o smooh shaes. Becase sbdivision can creae smooh crves and srfaces of arbirary oology, sbdivision is now an inegral ar of Geomeric Modeling []. The Lane-Riesenfeld sbdivision algorihm [6] is he mos commonly sed sbdivision algorihm for niform B-slines of arbirary degree. To sbdivide a B-sline crve of degree d, he algorihm roceeds in wo hases. The firs se dobles each conrol oin. This se is followed by d ronds of mid-oin averaging, where each edge is relaced by a verex locaed a he mid-oin of ha edge. Ieraing his sbdivision rocess generaes a seqence of iecewise linear crves ha in he limi converges o he niform B-sline defined by he original conrol oins. We can easily visalize his algorihm as he yramid shown in Figre. In his figre, arrows corresond o aking linear combinaions of he oins a he base of each arrow wih he weighs secified on each edge. This reeaed averaging aradigm is qie owerfl and lies a he fondaion of many srface sbdivision schemes based on niform B-slines of arbirary degree [7,0,,]. One of he disadvanages of he Lane-Riesenfeld algorihm is ha his algorihm only oeraes on B-slines wih niform kno sacing. Generally, however, B-slines can have non-niformly saced knos. Non-niform kno sacing forms he basis for NURBS crves and srfaces commonly sed in Comer Aided Design. Therefore, here we consider he qesion of wheher a sbdivision algorihm, similar o Lane-Riesenfeld sbdivision, can be consrced for non-niform B-slines. In ariclar, given a B-sline crve of degree d wih a monoonically increasing se of knos,,... n we wold like o doble he nmber of conrol oins by insering knos i sch ha i i i+ ; ha is, by
Non-niform Sbdivision for B-slines of Arbirary Degree + + + + + + + + + + + + + 8 8 8 8 + + + Figre. The Lane-Riesenfeld algorihm: he conrol oins are dobled (boom of he yramid) followed by d ronds of midoin averaging. Here we illsrae he cbic case. insering one new kno in each arameer inerval.. Previos Work There are acally several algorihms ha erform kno inserion on non-niform B- slines of arbirary degree. For examle, Boehm s kno inserion algorihm [] insers one kno a a ime. However, his rocedre ms be reeaed for every kno i, leading o a raher inefficien algorihm when comared o he Lane-Riesenfeld algorihm for niform kno sacing. The Oslo algorihm [] is similar o Boehm s algorihm exce ha he Oslo algorihm simlaneosly insers mlile knos beween wo of he original consecive knos. However, when we erform sbdivision, we wish o inser only one new kno beween each old air of consecive knos. In his seing he Oslo algorihm redces o Boehm s kno inserion algorihm. Barry e al. [] rovides a bonded deh varian of he Oslo algorihm for insering mlile knos. Similar o he Oslo algorihm, his algorihm excels when insering mlile knos beween each old air of consecive knos whereas we will only be insering a single new kno in each kno inerval. Sablonniere [9] inrodces a erahedral algorihm for erforming a change of basis beween fncions defined locally over wo differen kno seqences. This algorihm can be sed o erform kno inserion as well b is seldom sed in racice becase Sablonniere s algorihm is slower han boh Boehm s algorihm and he Oslo algorihm. Conribions We shall resen an efficien algorihm for sbdividing non-niform B-slines of arbirary degree in a manner similar o he Lane-Riesenfeld algorihm for niform B-slines. Or algorihm consiss of dobling he conrol oins followed by d ronds of non-niform averaging for a degree d B-sline crve where he averages deend on he local kno sacing of he crve. This algorihm has a srcre similar o he Lane-Riesenfeld algorihm,
Non-niform Sbdivision for B-slines of Arbirary Degree which is also comosed of dobling followed by d ronds of averaging; b in he Lane- Riesenfeld algorihm he averaging is niform and indeenden of he local kno sacing. Moreover, nlike he Lane-Riesenfeld algorihm where he sandard roof is based on he coninos convolion formla for he niform B-sline basis fncions [], or algorihm follows easily from blossoming [8].. Non-niform Sbdivision Consider a B-sline crve of degree d wih knos... n and conrol oins 0,..., n d. The conrol oin i is associaed wih he knos i+ i+... i+d and is reresened by he olar form or blossom of he B-sline crve evalaed a hese kno vales [8]. Ths we wrie i = b[ i+,..., i+d ]whereb is he blossom of he B-sline crve. Or goal is o inser new knos,,... n sch ha i i i+. Kno inserion is akin o sbdivision; or kno inserion rocedre will rodce a more highly refined conrol olygon relaive o he new kno seqence. As he knos become dense, his sbdivision rocedre will generae a seqence of iecewise linear crves ha converge in he limi o he non-niform B-sline crve for he original conrol oins. When all of he i are niformly saced and each i = i+ i+, hen he Lane-Riesenfeld algorihm is a simle, effecive rocedre for erforming his kno inserion. However, if he i or i are non-niform, hen more comlex algorihms ms be alied. While Boehm s algorihm can always be sed o erform kno inserion, Boehm s algorihm is slow and lacks he elegan srcre of he Lane-Riesenfeld algorihm. Or algorihm is comosed of wo ses: dobling followed by d ronds of averaging, where d is he degree of he crve. This srcre is idenical o he Lane-Riesenfeld algorihm, exce ha or averaging ronds deend locally on he kno srcre corresonding o he srronding conrol oins. b [,,,...] y x y x y x b [ x,,,...] b [ y,,,...],,,... x,,,... y,,,... Figre. The mli-affine roery of he blossom (lef) and a shor-hand noaion indicaing he insered kno where he weighs in he affine combinaion are imlici. The firs se in or algorihm, dobling, is exacly he same as he Lane-Riesenfeld algorihm and we simly doble he conrol oins in he crve. We hen follow his se wih d ronds of averaging. In he k h averaging se where k =,...,d, we locally inser he knos k, k, k, k+, k+,...sing he mli-affine roery of he blossom. This roery saes ha if b[x,,,...]andb[y,,,...] are conrol oins whose cor-
Non-niform Sbdivision for B-slines of Arbirary Degree 0 Figre. Or non-niform sbdivision algorihm for degree d = where conrol oins are referred o by he corresonding kno vales. The kno vale a he cener of each riangle indicaes he kno o inser locally sing he wo conrol oins a he base of he riangle. 0 Figre. Or non-niform sbdivision algorihm for degree d =. Noice ha he knos insered a level are idenical o hose sed in Figre. resonding kno seqences differ by a mos one vale (x and y in or examle) and he kno vale we wish o inser is, we simly form he affine combinaion b[,,,...]= y y x b[x,,,...]+ x y x b[y,,,...]. Figre deics his roery. On he lef he arrows indicae affine combinaions along wih he weighs associaed wih each edge. The righ side of he figre shows or eqivalen noaion ha secifies only he insered kno. Figre illsraes his algorihm for a linear B-sline. For convenience, we refer o he B-sline conrol oins simly by heir kno vales and omi he fncion call o he blossom b. The conrol oins si a he base of he yramid and are already dobled. The vale shown in he cener of each yramid indicaes he kno vale o inser sing he wo conrol oins a he base of he yramid. The resling conrol oins reresened by heir kno vales are shown a he o of he yramid. The resl of he comaion in Figre are he conrol oins for he desired kno seqence afer sbdividing he crve.
Non-niform Sbdivision for B-slines of Arbirary Degree 6 0 6 6 Figre. Or non-niform sbdivision algorihm for degree d =. The dashed edges indicae a weigh of zero for he verex a he base of he edge. Half of he old conrol oins are coied o he nex level a each rond of averaging. Figre deics or algorihm for a qadraic B-sline. The differen inensiies are designed o illsrae he nesing of he differen degrees. If we consider only he black ex, he algorihm is idenical o he algorihm in Figre for linear B-slines. The gray ex indicaes he differen knos and conrol oins needed for qadraic B-slines. For qadraic B-slines wo ronds of averaging are reqired o sbdivide he crve. A averaging se k = for he qadraic algorihm, he local kno vales ha we inser are idenical o hose from he linear algorihm. However, even hogh he kno vale is idenical, he affine combinaions ha we se a level k = for he qadraic algorihm are no idenical o he affine combinaions in he linear algorihm de o he differen kno seqences associaed wih he conrol oins. Ths nlike he Lane-Riesenfeld algorihm where he averaging rle is simly o ake he midoin a every level of averaging no maer wha he degree of he crve, he averaging rles in or algorihm deend on he degree of he crve. Neverheless, afer wo ronds of averaging he kno seqence is exacly ha of he sbdivided crve. Figre shows or algorihm for a cbic B-sline crve wih he gray-levels designed o show he nesing of he algorihm for differen degree B-slines. Noice ha every oher edge in he yramid is dashed. For hese yramids he kno insered is already resen in one of he wo conrol oins a he boom of he yramid. Therefore, hese conrol oins are simly romoed o he nex level wiho erforming any affine combinaion. The fac ha hese conrol oins are simly coied o he nex level illsraes ha or algorihm does no redce o he Lane-Riesenfeld algorihm even when he original knos i are niformly saced and he new knos i = i+ i+. Noice hen ha or algorihm erforms only half as many mlilicaions and addiions for one rond of sbdivision as Lane-Riesenfeld sbdivision hogh he laer ses only combinaions of.
Non-niform Sbdivision for B-slines of Arbirary Degree 6 We can also wrie he oeraions in or algorihm comacly as a recrrence involving he conrol oins and he kno vales. Le i be he conrol oins of he B-sline crve associaed wih he knos i+ i+... i+d,whered is he degree of he crve. The oins k i a he k h level of or algorihm are hen given by 0 i = i 0 i+ = i k i = i+ k i+d+ i+k i+d+ i+(k+)/ i+ k + i+k i+(k+)/ i+d+ i+(k+)/ k (i+) k i+ = i+d+ i+k i+d+ i+k/ k i+ + i+k i+k/ i+d+ i+k/ k (i+) k is odd k is even for k =,...,d.theoins d i reresen he resl of one rond of sbdivision. Finally, noice ha or sbdivision scheme will rodce a seqence of iecewise linear crves ha converge in he limi o he non-niform B-sline crve reresened by he original conrol oins and knos rovided ha he new knos i insered a each level of sbdivision lead o a dense covering of arameer sace. Figre 6 shows an examle of sbdivision for a non-niform B-sline basis fncion of degree sing or algorihm where he kno sacing is niform on boh he lef and righ sides of he basis fncion, b he knos are wice as dense on he lef as on he righ. In his examle, we simly inser he new knos i a he midoin of each consecive air of old knos.. Conclsions and Fre Work Or algorihm is a simle, fas mehod for erforming sbdivision on non-niform B-slines of arbirary degree. Similar o he Lane-Riesenfeld algorihm, or echniqe is comosed of dobling he conrol oins followed by d non-niform averaging ses where d is he degree of he crve. Or algorihm does no redce o he Lane-Riesenfeld algorihm when he original knos are niformly saced and he new knos lie a he midoins of he old knos. However, or mehod erforms only half as many comaions as he Lane-Riesenfeld algorihm. Moreover or mehod is easier o derive han he Lane- Riesenfeld algorihm. The sandard roof of he Lane-Riesenfeld algorihm is based on he observaion ha he basis fncions for niform B-slines can be generaed by coninos convolion []. In conras, he roof of or algorihm is immediae from he dal fncional and mliaffine roeries of he blossom. Blossoming roofs of he Lane- Riesenfeld algorihm are also ossible, b hese roofs are no so simle []. In he fre, we wold like o exlore wheher or no we can consrc an algorihm ha mimics he Lane-Riesenfeld algorihm even closer. Or crren echniqe reqires ha we know he degree of he crve before sbdivision, since he degree inflences he affine combinaions sed in he local kno inserion se. Ideally, we cold erform hese averaging asses wiho knowing he degree of he crve ahead of ime, hogh crren exerimens indicae ha his ye of algorihm may no be ossible in he flly nonniform seing []. Neverheless, if we lace resricions on he original knos i or he new knos i, sch an algorihm may exis. We also believe ha here exiss an algorihm for each ariclar degree ha redces o he Lane-Riesenfeld algorihm wiho any
Non-niform Sbdivision for B-slines of Arbirary Degree 7 Figre 6. Sbdivision of a non-niform cbic B-sline basis fncion sing or echniqe. resricions on he knos. However, hese algorihms are secial in each degree; he rles from one degree o anoher are comleely differen and, hence, do no nes like or algorihm or he Lane-Riesenfeld algorihm. Aendix A rovides wo examles of nonniform kno inserion algorihms for cbic B-slines ha redce o he Lane-Riesenfeld algorihm for cbic B-slines when he knos are evenly saced. We wold also like o generalize or mehod o creae a qadrilaeral sbdivision scheme for non-niform B-sline srfaces similar o he way he Lane-Riesenfeld algorihm is sed o creae srface sbdivision schemes [7,0,,]. The ensor-rodc case is relaively sraighforward and does no ose a challenge. However, exraordinary verices comlicae he algorihm. One of he bigges roblems is simly encoding he kno vecors for he verices in he resence of exraordinary verices. REFERENCES. Philli Barry and Ri-Feng Zh. Anoher kno inserion algorihm for B-sline crves. Comer Aided Geomeric Design, 9():7 8, 99.. W. Boehm. Insering new knos ino B-sline crves. Comer Aided Design, ():99 0, 980.. Thomas J. Cashman, Neil A. Dodgson, and Malcolm A. Sabin. Non-niform B- Sline sbdivision sing refine and smooh. In IMA Conference on he Mahemaics of Srfaces, ages 7, 007.. E. Cohen, T. Lyche, and R. Riesenfeld. Discree B-slines and sbdivision echniqes in comer-aided geomeric design and comer grahics. Comer Grahics and Image Processing, :87, 980.. Tony DeRose, Michael Kass, and Tien Trong. Sbdivision srfaces in characer animaion. In SIGGRAPH 98: Proceedings of he h annal conference on Comer grahics and ineracive echniqes, ages 8 9, 998. 6. J. Lane and R. Riesenfeld. A heoreical develomen for he comer generaion and dislay of iecewise olynomial srfaces. IEEE Transacions on Paern Analysis and Machine Inelligence, : 6, 980. 7. Harm Prazsch. Smoohness of sbdivision srfaces a exraordinary oins. Adv. in Com. Mah., 9: 77-90, 998. 8. L. Ramshaw. Blossoms are olar forms. Comer Aided Geomeric Design, 6: 8, 989. 9. P. Sablonniere. Sline and Bezier olygons associaed wih a olynomial sline crve. Comer Aided Design, 0():7 6, 978.
Non-niform Sbdivision for B-slines of Arbirary Degree 8 Figre 7. A sbdivision algorihm for non-niform cbic B-slines ha redces o he Lane-Riesenfeld algorihm when he knos are niformly sace. 0. Jos Sam. On sbdivision schemes generalizing niform B-sline srfaces of arbirary degree. Comer Aided Geomeric Design, 8():8 96, 00.. E. Voga and R. Goldman. Two blossoming roofs of he Lane-Riesenfeld algorihm. Coming, 79: 6, 007.. Joe Warren and Henrik Weimer. Sbdivision Mehods for Geomeric Design: A Consrcive Aroach. Morgan Kafmann Pblishers Inc., 00.. Denis Zorin and Peer Schröder. A nified framework for rimal/dal qadrilaeral sbdivision schemes. Comer Aided Geomeric Design, 8():9, 00. A. Non-niform Sbdivision Redcing o he Lane-Riesenfeld Algorihm Or algorihm for insering knos ino non-niform B-slines of arbirary degree does no redce o he Lane-Riesenfeld algorihm when he knos are niformly saced. However, for a ariclar degree, we believe ha i is always ossible o consrc a kno inserion algorihm ha redces o he Lane-Riesenfeld algorihm when he knos are evenly saced. Figre 7 shows an examle of sch an algorihm for cbic B-slines. This algorihm reqires he inserion of sedo-knos ha are no ar of he original or final kno seqences. However, hese knos case he algorihm o redce o he Lane- Riesenfeld algorihm when he original knos i are niformly saced and he new knos i = i+ i+. The disadvanage of his cbic algorihm is ha i does no generalize o B-slines of differen degrees. Frhermore, he algorihm iself is no niqe; here exiss many differen sedo-knos and inserion orders ha will rodce he same kno seqence afer
Non-niform Sbdivision for B-slines of Arbirary Degree 9 Figre 8. Anoher sbdivision algorihm for non-niform cbic B-slines ha redces o he Lane-Riesenfeld algorihm when he knos are niformly sace. sbdivision and redce o he Lane-Riesenfeld algorihm for niform knos. For examle, Figre 8 illsraes anoher sbdivision algorihm for non-niform cbic B-slines ha redces o he Lane-Riesenfeld algorihm when he knos are eqally saced. Again, his algorihm relies on he inserion of sedo-knos b does no rodce ha same inermediae affine combinaions. Crrenly, i is an oen roblem wheher or no here exiss an algorihm ha redces o he Lane-Riesenfeld algorihm for niform knos and has a simle, nesed srcre relaive o he degree of he B-sline crve.