Bicubic Polar Subdivision

Transcription

1 Bicubic Polar Subdivisio K. Karčiauskas Vilius Uiversity ad J. Peters Uiversity of Florida We describe ad aalyze a subdivisio scheme that geeralizes bicubic splie subdivisio to cotrol ets with polar structure. Such cotrol ets appear aturally for surfaces with the combiatorial structure of objects of revolutio ad at poits of high valece i subdivisio meshes. The resultig surfaces are C 2 except at a fiite umber of isolated poits where the surface is C ad the curvature is bouded. Categories ad Subject Descriptors: I.3.5 []: Computatioal Geometry ad Object Modelig; J.6 []: Computer-Aided Egieerig Geeral Terms: Algorithms Additioal Key Words ad Phrases: Subdivisio, polar layout, polar et, bicubic, Catmull-Clark, curvature cotiuity. INTRODUCTION Polar cotrol ets (Figure ) capture the combiatorial structure of objects of revolutio ad are therefore more atural at poits of high valece (see e.g. Figure 2) tha the all-quads layout favored by Catmull-Clark subdivisio [Catmull ad Clark 978]. Correspodigly, we defie ad aalyze i the followig a biary subdivisio scheme that, just like Catmull-Clark subdivisio, geeralizes the refiemet rules of uiform cubic splies but for the layout of a polar et. Formally, a cotrol et without boudary is a polar et [Karčiauskas ad Peters 2007] if it cosists of extraordiary mesh odes surrouded by triagles, ad of quadrilaterals that have odes of valece four. The extraordiary mesh odes eed oly be separated by oe layer of odes of valece four as illustrated i Figure 7, left. Applyig quad-tri subdivisio [Stam ad Loop 2003; Peters ad Shiue 2004; Schaefer ad Warre 2005] to a polar et is ot a good alterative, sice Loop subdivisio also i 2 c i, c i,2 radial i+ A circular Fig.. Polar cotrol et ear a extraordiary poit (left) ad its refiemet (right) uder subdivisio. The cotrol poits c ij have subscripts i idicatig (modulo the valece ) the directio ad subscripts j idicatig the radial distace to the extraordiary poit c i0. Oly the radial, ot the circular directio is refied. does ot cope well with such iput meshes (Figure 2). Polar subdivisio differs structurally from tesored uivariate schemes with sigularities, e.g. [Mori et al. 200], i that the umber of eighbors of the extraordiary poit does Permissio to make digital/hard copy of all or part of this material without fee for persoal or classroom use provided that the copies are ot made or distributed for profit or commercial advatage, the ACM copyright/server otice, the title of the publicatio, ad its date appear, ad otice is give that copyig is by permissio of the ACM, Ic. To copy otherwise, to republish, to post o servers, or to redistribute to lists requires prior specific permissio ad/or a fee. c 20YY ACM /20YY/ $5.00 ACM Trasactios o Graphics, Vol. V, No. N, Moth 20YY, Pages 0??.

2 2 K. Karčiauskas ad J. Peters Fig. 2. Wrikle removal o a Easter Islad head (valece 20). (from left to right) Catmull-Clark subdivisio, Loop subdivisio (quad facets are split), cotrol et, color-coded rigs of the polar subdivisio surface, polar subdivisio surface. ot double with each polar subdivisio step but stays fixed. Quadrilaterals i a polar et are ot split ad the cotrol et refies oly towards the extraordiary poit (Figure ). Therefore, the polar cotrol et does ot, off had, serve the fuctio of approximatig the surface ever more closely by smaller facets. However, the resultig surface as a sigle B-splie mesh growig towards the extraordiary poit, i.e. the surface does ot have the cascadig sequece of T-corers itrisic to Catmull-Clark surfaces. Sectio 4, Cotrol Nets, explais this i detail. Compared to [Karčiauskas et al. 2006], the more localized computatio of bicubic polar subdivisio results i a more localized curvature distributio. At the extraordiary poit, the curvature of surfaces geerated by bicubic polar subdivisio is oly bouded but eed ot be cotiuous while the algorithm i [Karčiauskas et al. 2006] geerates C 2 surfaces. The preset scheme has, however, the advatage of simpler rules without visibly sacrificig good shape. 2. POLAR REFINEMENT RULES Apart from the extraordiary mesh odes, the polar et defied i the itroductio, is a stadard bicubic B-splie cotrol et. For the layer of quadrilaterals adjacet to the triagles, we iterpret the triagles as degeerate quadrilaterals with oe edge collapsed. It is easy to check, for example by coversio to Bézier form, that this iterpretatio does ot result i sigularities i the quadrilateral layer. I order to map a polar et to a refied polar et, we will refie the bicubic splie et oly i the radial directio (cf. Figure ). γ i+ β γ i γ i Fig. 3. Refiemet stecils for biary polar subdivisio. As is typical for subdivisio algorithms, we eed oly explai how to refie the polar et immediately coected to extraordiary mesh odes. To obtai leadig eigevalues, 2, 2, 4, 4, 4, it suffices to have special rules oly at the extraordiary mesh ode ad its direct eighbors (Figure 3). The two regular rules are the subdivisio rules for ACM Trasactios o Graphics, Vol. V, No. N, Moth 20YY.

3 uivariate uiform cubic splies. The extraordiary rules have the weights := β 4, β := 2, γ k := Bicubic Polar Subdivisio 3 ( β ck + (ck )2 + 2 (ck )3), c k := cos( 2πk). () Here, we chose β = /2 to emphasize covexity at the extraordiary poit, sice this is likely the domiat sceario for polar meshes. This choice is also reasoable for saddles (Figure 9). Sectio 4, Covexity ad Valece discusses the role of β i more detail. A useful property of polar surfaces is that the valece ca be chaged by iterpretig each circular rig of coefficiets as the cotrol polygo of a cubic splie curve. To avoid a special discussio of low valeces, we uiformly isert kots i the circular splie curves ad double the valece whe {3, 4, 5}. That is, we may assume 6 i the followig. 3. PROPERTIES BY CONSTRUCTION Let c m i,j be the cotrol poit of the ith sector ad the jth layer as idicated i Figure. The cetral ode is cosidered split ito copies c m i0, each weighted by /. The the vector of cotrol poits c m := (...,c m i0,c m i,c m i2,c m i3,...) R 4 4, is refied by a subdivisio matrix with block-circulat structure: c m+ = Ac m, A := A 0 A... A A A 0... A A... A A 0 R 4 4, A 0 := 0 0 β γ , A i := that ca be block-diagoalized by Discrete Fourier Trasform Âi := the eige-aalysis is pleasatly simple. k=0 ωik 0 0 β γi A k,, i =,...,, ω l := exp( 2πl ), so that LEMMA. For geeric iput data, the limit surface of bicubic polar subdivisio is C 2 except at isolated extraordiary poits where the surface is C ad the curvature bouded. PROOF. As illustrated i Figure 4, cotrol poit layers through 5 defie two rigs of bicubic splies (Figure 4 middle). This double-rig is C 2 sice it correspods to a regular (periodic) tesor-product splie. As i Catmull-Clark subdivisio, cosecutive double-rigs joi C 2. For > 5, 2, if i {, } ( 0 0 ) ( ) ˆγ i = 4, if i {2, 2}, ad 6, if i {3, 3} Â0 = 0, if i > 3 ad i < 3. β β ,  i = 0 ˆγ i (2) The eigevalues of Â0 are, 4, 8, 0 ad the eigevalues of Âi for i =,...,, are ˆγ i := k=0 ωik γ k, 8, 0, 0. I particular, λ = ˆγ ad (λ ) 2 = λ 2 = ˆγ 2 as is required for bouded curvature. Sice the eigevector of matrix  for λ = 2 is (0,, 2, 3)t, the subdomiat eigevectors of A are the coordiates of [ ] v = (...,r3 i,r i 0,r i,r i 2,r i 3,r i+ 0,...), r i cos i 2π k := k, i =,...,, k = 0,, 2, 3. (3) si i 2π The cotrol et v defies the characteristic map (Figure 4, middle) [Reif 995], whose regularity ad ijectivity are easily verified [Peters ad Reif 998; Umlauf 999]. The eigevectors correspodig to the eigevalue /4 are from Fourier blocks 0, 2 ad 2 ad they are ot geeralized eigevectors. Explicitly, for use i Sectio 4, the ACM Trasactios o Graphics, Vol. V, No. N, Moth 20YY.

4 4 K. Karčiauskas ad J. Peters Fig. 4. (left) Layers 0 through 5 (geerated by oe subdivisio of layers 0 through 3) defie (middle) oe piecewise bicubic double-rig. (right) Cosecutive double-rigs joi smoothly ad, ulike Catmull-Clark subdivisio, without T-corers. eigevectors v 2k to the eigevalue 4 of Âk for Fourier idex k {0, 2} are v 20 := ( + 3b,, 7 + 3b, 6 + 6b) t, b := 4β, v 22 := (, 4, 6, 4) t. (4) Together with the curvature bouded spectrum, this implies curvature boudedess as claimed. LEMMA 2. The limit extraordiary poit is ηc 00 + ( η) c i η := i= 4( β). 3 PROOF. We choose the represetatio Ā R3+ 3+ of the subdivisio operator where we do ot replicate the cetral ode c 00 : a r... a r a c Ā 0... Ā a Ā := r := [ (...., 0, 0] γ0 0 0 ) (. a c := [ β, 3 8, Ā 0 := 0]t γi ) 0 0 Ā i := 0 0 0, i =,..., a c... Ā Ā 0 We ca directly check that the left eigevector of Ā with respect to the domiat eigevalue is β [ β +, l, l,...,l]t, l := [, 0, 0]. ( β + ) The claim follows (see [DeRose et al. 998], Appedix A) sice the etries sum to. Sice [0,, 0, 0] is a left eigevector to Â, the ormal directio at the extraordiary poit is simply ( i= cosi2π c i0) ( i= si i2π c i0). 4. DISCUSSION This sectio discusses some alterative schemes, the meaig of cotrol polyhedra ad adjustmet of valece ad covexity. Alterative Schemes. The bicubic polar subdivisio algorithm has special rules for both the ew cetral ode ad its direct eighbors. Choosig symmetric special rules oly for the cetral ode does ot yield appropriate degrees of freedom for smoothess. Specifically, forcig a double subdomiat eigevalue by tuig oly the rules for the cetral ode, leads to oe sigle subdomiat eigevector for > 3; oly for = 3, do there exist rules to geerate C surfaces with a spectrum suitable for bouded curvature. So, a direct polar aalogue of Catmull-Clark subdivisio fails ad the questio arises whether a terary polar subdivisio scheme, aalogous to [Loop 2002], is advatageous. ACM Trasactios o Graphics, Vol. V, No. N, Moth 20YY

5 Bicubic Polar Subdivisio 5 β γ i+ γ i γ i Fig. 5. Refiemet stecils for a terary polar subdivisio (splittig ito three i the radial directio) where β := 38 8, := β 9, γ k := 4 8 `5 + 2c k ` + c k 2. We derived such a variat for compariso (see Figure 5). The weights γ k are o-egative ad the scheme satisfies all the costraits o the leadig eigevalues (, 3, 3, 9, 9, 9 ) ad eigevectors for curvature boudedess. However, the resultig surfaces did ot look better tha those of the proposed biary subdivisio. Cotrol Nets ad Surface Rigs. Subdivisio surfaces ca either be viewed as refiig a cotrol et or as geeratig a sequece of surface rigs covergig to the extraordiary poit [Reif 995]. The first serves ituitio if the cotrol et outlies the shape, the secod is preferred for exact evaluatio, computig ad aalysis. Both Catmull-Clark subdivisio ad polar subdivisio admit the two views but differ i their bias. To see this, defie a T-corer to be the locatio where a edge betwee two distict polyomial patches meets the midpoit of a edge of a third. With each refiemet, Catmull-Clark subdivisio geerates T-corers betwee the patches of adjacet surface rigs (Figure 6, left top). Polar subdivisio does ot geerate T-corers (Figure 6, right top) sice the cotrol et refies oly towards the extraordiary poit (Figure ). Oe approach for geeratig a faceted approximatio covergig to the uderlyig surface is to split the quadrilaterals of the polar et at each refiemet ito four ad leave the triagles utouched. This yields T-corers i the faceted approximatio (Figure 6, right bottom). Reflectig the bias towards presetig a mesh without T- corers versus obtaiig a surface without T-corers, Catmull-Clark subdivisio is usually illustrated by a sequece of cotrol ets (Figure,left bottom), hidig the surface T-corers, while polar surfaces are preferably itroduced as a sequece of surface rigs. patches cotrol facets Catmull-Clark polar Fig. 6. Layout of patches ad cotrol polyhedro for Catmull-Clark subdivisio (left) ad polar subdivisio (right). The T- corers i Catmull-Clark (left top) are itrisic (the coarser patch is C at the T- corer). The T-corers i the refied polyhdero (right bottom) are optioal ad ot part of the polar et. Covexity ad Valece. Decreasig the parameter β i () pulls the surface closer to the extraordiary mesh ode. Recetly [Gikel ad Umlauf 2006] documeted how such straightforward maipulatio results i a limit surface i the desirable regio of a shape chart [Karciauskas et al. 2004]: decreasig β emphasizes covexity. We therefore chose β := /2 (see Figure 0) over β := 5/8 eve though the latter yields o-egative weights γ k = 8 (+ck )(+2ck )2 0. Table I shows the effect of β o the subsubdomiat eigevector v 20 of (4), that determies the shape i the covex settig, ad its secod differece v 20. For β = /2, the sector partitio curves are quadratic ad have a more proouced curvature tha for β = 5/8. We also observed that icreasig the valece by kot isertio improves the curvature distributio for covex eighborhoods (see e.g. Figure 0). This is due to the icreased symmetry of v 20 ad the fact that, if a curve is C at the cetral poit ad opposite curve segmets are mirror images, the the curve is C 2. ACM Trasactios o Graphics, Vol. V, No. N, Moth 20YY.

6 6 K. Karčiauskas ad J. Peters β 3/8 4/8 5/8 6/8 v 20 (, 5, 29, 68) t (, 2,, 26) t (,,5, 2) t (, /2, 2, 5) t 2 v 20 (8, 5) t (6, 6) t (2, 3) t (0, 3/2) t 5. CONCLUSION Table I. Coefficiets of v 20, the eigevector of the zeroth Fourier mode to the eigevalue /4. The algorithm just defied ad aalyzed is a polar cousi of Catmull-Clark subdivisio. Its curvatures are bouded just as [Sabi 99]. Its simplicity ad the fact that the output cosists of bicubic patches recommed bicubic polar subdivisio as a useful additio to Catmull-Clark subdivisio. This additio gives the desiger more freedom just where Catmull-Clark subdivisio ecouters shape deficiecies, the valece is high or polar layout is atural. The paper [Myles et al. 2007] explais i detail how bicubic Catmull-Clark ad bicubic polar subdivisio ca be combied for smooth object desig such as i Figure 2. ACKNOWLEDGMENTS This work was supported i part by NSF Grats DMI ad CCF Ashish Myles implemeted the algorithm. REFERENCES CATMULL, E. AND CLARK, J Recursively geerated B-splie surfaces o arbitrary topological meshes. Computer Aided Desig 0, DEROSE, T., KASS, M., AND TRUONG, T Subdivisio surfaces i character aimatio. I Siggraph 998, Computer Graphics Proceedigs, M. Cohe, Ed. ACM Press, GINKEL, I. AND UMLAUF, G Loop subdivisio with curvature cotrol. I Proceedigs of Symposium of Graphics Processig (SGP), Jue , Cagliari, Italy, A. Scheffer ad K. Polthier, Eds. ACM Press, KARČIAUSKAS, K., MYLES, A., AND PETERS, J A C 2 polar jet subdivisio. I Proceedigs of Symposium of Graphics Processig (SGP), Jue , Cagliari, Italy, A. Scheffer ad K. Polthier, Eds. ACM Press, KARČIAUSKAS, K. AND PETERS, J Surfaces with polar structure. Computig 79, KARCIAUSKAS, K., PETERS, J., AND REIF, U Shape characterizatio of subdivisio surfaces case studies. Computer-Aided Geometric Desig 2, 6 (july), LOOP, C Smooth terary subdivisio of triagle meshes. I Curve ad Surface Fittig, Sait-Malo. Vol. 0(6). Nashboro Press, 3 6. MORIN, G., WARREN, J., AND WEIMER, H A subdivisio scheme for surfaces of revolutio. Comp Aided Geom Desig 8, 5, MYLES, A., KARČIAUSKAS, K., AND PETERS, J Extedig Catmull-Clark subdivisio ad PCCM with Polar structures. I Proceedigs of Pacific Graphics, Hawaii, M. Alexa, S. Gortler, ad T. Ju, Eds. ACM Press, xx xx. PETERS, J. AND REIF, U Aalysis of geeralized B-splie subdivisio algorithms. SIAM Joural o Numerical Aalysis 35, 2 (Apr.), PETERS, J. AND SHIUE, L.-J Combiig 4- ad 3-directio subdivisio. ACM Tras. Graph 23, 4, REIF, U A uified approach to subdivisio algorithms ear extraordiary vertices. Comp Aided Geom Desig 2, SABIN, M. 99. Cubic recursive divisio with bouded curvature. I Curves ad Surfaces, L. S. P.J. Lauret, A. LeMéhauté, Ed. Academic Press, SCHAEFER, S. AND WARREN, J. D O C 2 triagle/quad subdivisio. ACM Tras. Graph 24,, STAM, J. AND LOOP, C. T Quad/triagle subdivisio. Comput. Graph. Forum 22,, UMLAUF, G Glatte freiformfläche ud optimierte uterteilugsalgorithme. Ph.D. thesis, Computer Sciece. ACM Trasactios o Graphics, Vol. V, No. N, Moth 20YY.

7 Bicubic Polar Subdivisio 7 Fig. 7. Mirrored 6-sided pyramid. (left) cotrol et; (middle ad right) subdivisio surfaces (three light sources) usig (middle) Catmull-Clark subdivisio, (right) polar bicubic subdivisio (with Gauss-curvature shaded iset). Fig. 8. Iteded ripples: (left) Cotrol et (middle) Catmull-Clark subdivisio (ote additioal micro-ripples); (right) polar subdivisio. Fig. 9. Nocovex polar et, ested surface rigs of polar subdivisio, shaded surface ad reflectio lies o the saddle. cotrol et = 6, β = 5 8 = 2, β = 5 8 = 6, β = 2 = 2, β = 2 Fig. 0. The effect of chagig the parameters β ad of bicubic polar subdivisio o the everywhere positive Gauss-curvature of a capped cylider. ACM Trasactios o Graphics, Vol. V, No. N, Moth 20YY.

8 8 K. Karčiauskas ad J. Peters Fig.. (left) Sample meshes; (middle) bicubic surface rigs; (right) Polar subdivisio surfaces. Fig. 2. Bicubic subdivisio with Catmull-Clark rules applied where 4 quadrilaterals meet ad polar rules where triagles meet (grey surfaces). ACM Trasactios o Graphics, Vol. V, No. N, Moth 20YY.