Improved triangular subdivision schemes 1

Transcription

1 Improved triagular subdivisio schemes Hartmut Prautzsch 2 Georg Umlauf 3 Faultät für Iformati, Uiversität Karlsruhe, D-762 Karlsruhe, Germay 2 prau@ira.ua.de 3 umlauf@ira.ua.de Abstract I this article we improve the butterfly ad Loop s algorithm. As a result we obtai subdivisio algorithms for triagular ets which ca be used to geerate G - ad G 2 - surfaces, respectively. Keywords: Subdivisio, iterpolatory subdivisio, Loop s algorithm, butterfly algorithm.. Itroductio Subdivisio algorithms are popular i CAGD sice they provide simple, efficiet tools to geerate arbitrary free form surfaces. For example, the algorithms by Catmull ad Clar [3] ad Loop [7] are geeralizatios of well-ow splie subdivisio schemes. Therefore the surfaces produced by these algorithms are piecewise polyomial ad at ordiary poits curvature cotiuous. At extraordiary poits however, the curvature is zero or ifiite. I geeral, sigularities at extraordiary poits is a iheret pheomeo of subdivisio, see [3, 2, 9]. The smoothess of a subdivisio surface at its extraordiary poits depeds o the spectral properties of the associated subdivisio matrix. Doo ad Sabi [4] derived ecessary coditios o the eigevalues. Ball ad Storry [, 2] made first rigorous ivestigatios to prove the taget plae cotiuity for a class of Catmull/Clar type algorithms. The Reif [] observed that taget plae cotiuous surfaces may have local selfitersectios ad itroduced the characteristic map defied by the subdomiat eigevectors. Moreover, for all statioary subdivisio schemes he derived ecessary ad sufficiet coditios which guaratee that the limitig surface is regular, i.e. taget plae cotiuous without local peetratios. Fially, i [] Reif s characteristic map is used to parametrize the subdivisio surface. With this parametrizatio Supported by DFG grat # PR 565/- it is possible to extet Reif s result ad to obtai for all statioary subdivisio schemes ecessary ad sufficiet coditios which guaratee that the limitig surface is a regular G -surface. Doo ad Sabi [4], Ball ad Storry [2] ad Loop [7] used the smoothess criteria to fid amog certai variatios of the Catmull/Clar ad Loop s algorithm the best. However, these best algorithms still produce curvature discotiuous surfaces, see e.g. [2]. I [] we too a differet approach. Istead of varyig the subdivisio rules withi some bouds which are set heuristically, we chaged the spectrum of the subdivisio matrix so as to obtai the desired properties. Usig the G 2 -characterizatio i [] we derived a G 2 -subdivisio algorithm from the Catmull/Clar algorithm (which does ot produce ifiite curvatures), see []. Here we provide similar improvemets, a G - ad a G 2 - algorithm based o the butterfly ad Loop s algorithm. 2. Loop s algorithm Loop s algorithm geeralizes the subdivisio algorithm for surfaces expressed i terms of the symmetric quartic box splie over a regular triagulatio of IR 2. It geerates from ay triagular et N a ew et N, whose vertices are classified as E- ad V-vertices. Computig the weighted averages of the four vertices of ay two triagles i N sharig a commo edge with the weights show i Figure gives the E-vertices. Similarly computig the weighted averages of all vertices of all triagles i N aroud ay vertex with the weights show i Figure gives the V-vertices. For = 6 Loop chooses (6) = 5= sice this correspods to box splie subdivisio. The ew et N is obtaied by coectig for all triagles of N the associated three E-vertices ad for all edges of N the associated E-vertices with both associated V-vertices. By the same procedure a ext et N 2 is obtaied from N ad so o.

2 3 E-mas V-mas Figure. The mass of the Loop algorithm the V-mas is illustrated for =6. Note that a vertex of ay et N i i is extraordiary, i.e. a iterior vertex with valece 6= 6, if it is a V-vertex associated with a extraordiary vertex of N i;. Thus the umber of extraordiary vertices is costat for all ets N i i ad these vertices are separated by more ad more ordiary vertices as i grows. I particular if N is a regular triagular et, i.e. without extraordiary vertices, Loop s algorithm coicides with the subdivisio algorithm for quartic box splie surfaces. Thus also for a arbitrary et N the sequece N i coverges to a piecewise quartic surface with oe extraordiary poit for each extraordiary vertex of N. The limitig surface is a C 2 -surface everywhere except at its extraordiary poits. Loop s aalysis shows that the limitig surface has a cotiuous taget plae at its extraordiary poits for a certai rage of s, see [7]. 3. The butterfly algorithm The butterfly algorithm of Dy et al. [5] geerates a sequece of triagular ets N i i similar to Loop s algorithm. Oly the mass used to compute the E- ad V- vertices are differet. They are give i Figure 2. A sequece of ets N i obtaied by the butterfly algorithm with small positive! coverges to a surface that is differetiable everywhere except at its extraordiary poits of valece 3 [5, 6] ad. At extraordiary poits of valece the surface is taget plae cotiuous but it has self-itersectios ad therefore is ot regular. We checed this for several!. However, i the sequel we always wor with! ==32. Variatios of the butterfly algorithm have bee proposed by Zori et al. [6]. However, the smoothess of the limitig surfaces obtaied by these variatios has ot yet bee ivestigated. 4. A smoothess coditio I Sectios 5 ad 6 we preset modificatios of Loop s ad the butterfly algorithm givig G 2 -org -surfaces i the limit. The method used to derive these modificatios is based o the G -aalysis of subdivisio schemes give i [] ad ca also be used for subdivisio schemes for quadrilateral ets []. For more details we eed to recall a result from []. We preset it i the theorem below for ay subdivisio scheme S that is idetical with the butterfly or Loop s algorithm except that E- ad V-mass may be differet. We assume that the limitig surface associated with ay iitial triagular et N obtaied by the subdivisio scheme S has C -parametrizatios aroud all its ordiary poits. Extraordiary poits are isolated as observed i Sectio 2. Therefore, to aalyze the smoothess of the limitig surface at extraordiary poits it suffices to cosider a subet M of N cosistig of oe extraordiary vertex surrouded by say r rigs of ordiary vertices as illustrated i Figure 3 for r =3. 2ω 2 2 2ω E-mas V-mas Figure 2. The mass of the butterfly algorithm. Figure 3. A et with oe extraordiary vertex of valece 5 (mared by ) surrouded by r =3rigs of ordiary vertices.

3 Further let M be the largest subet of N whose vertices deped oly o M. This et M also has oly oe extraordiary vertex surrouded by say r rigs of ordiary vertices. Note that r is roughly twice as lage as r. For example i Loop s algorithm r = 5if r = 3ad i the butterfly algorithm r =6if r =4. Let r be so large that r ; r. The discardig the r ; r outer rigs of M gives a et K with the same size ad coectedess as M. Let m ::: m m ad ::: m deote the vertices of M ad K, respectively. Sice the vertices i are affie combiatios of the m j, there is a m m matrix A such that [ ::: m ] t = A[m ::: m m ] t : Let s deote the limitig surface associated with M uder the subdivisio scheme S. Applyig S to M gives the same limitig surface s, but the surface s associated with the subet K is smaller ad oly a part of s. Taig s away from s gives the here so-called first surface rig associated with M. Now we are able to preset the followig theorem which is prove i more geeral form i []: Theorem 4. Let A have the m (possibly complex) eigevalues :::, where > jj jj ::: jj ad assume two eigevectors c ad d associated with the double eigevalue. If the first surface rig of the et give by [c :::c m ] t =[cd] is regular without self-itersectios ad jj > jj (4.) the the limitig surface is a G -surface for almost all iitial ets M. (More precisely, the limitig surface is a G -surface for all iitial ets M whose expasio by the eigevectors of A ivolves c i oe ad d i a secod coordiate.) The eigevalue coditio (4.) goes bac to Doo ad Sabi [4]. The first surface rig associated with the eigevectors c ad d is called the characteristic map of A by Reif who used it to prove this Theorem for =[]. If the limitig surface i Theorem 4. is a C -maifold, 2, the the extraordiary poit is a flat poit. This fact is also true for more geeral subdivisio schemes, see [, 9]. of Loop s algorithm satisfyig the G 2 -coditio we diagoalize the matrix A, A = V V ; chage the modal matrix to = diag( ::: ) where =diag( ::: ) ad compute the ew subdivisio matrix as A = V V ; : where j j ::: j j < 2 Lemma 5. The matrices A ad A have the same characteristic maps. Proof The eigevectors associated with are the same for A ad A. They defie a plaar cotrol et N. Subdividig N by Loop s algorithm ad also by the modificatio results both times i the same sequece of ets N i. The extraordiary vertex ad its three surroudig rigs of cotrol poits i N i are scaled versios of N. The other cotrol poits of N i are computed by the subdivisio rules for regular ets. Thus Loop s algorithm ad its modificatio applied to N produce the same surface i the limit. 2 The symmetry of Loop s scheme meas that the subdivisio matrix A is bloc-circulat. Therefore a discrete Fourier trasformatio ca be used to aalyze the spectral properties of A. If =3, the matrix A has the subdomiat eigevalue = =4 ad exactly six eigevalues with modulus i the half-ope iterval [jj 2 jj). These are the two triple eigevalues = ad =6. Chagig just these triple eigevalues to the triple eigevalues = +" ad =6 + " 2, respectively, such that = +" ad =6 + " 2 are less tha jj 2, results i a matrix A, which represets the same mass as the origial matrix A except for the E- ad V-mass show i Figure 4, where Modificatios of Loop s algorithm The subdivisio matrix A of Loop s algorithm associated with a extraordiary vertex of valece has a sigle domiat eigevalue ad satisfies the G -coditios of Theorem 4. [7, 5], but ot the G 2 -coditio [4]. To obtai a subdivisio matrix A that represets a modificatio E-mas V-mas Figure 4. The E- ad V-mass of the modified Loop algorithm ear a vertex of valece = 3.

4 = + " 2(2;) = 3 ; (6;7)" 6(2;) 2 = + " 3 = (2;)" 6(2;) = 6 + " 2; ; 3"2 6;7 = 5 ; (6;7)" 3(2;) ; (6;)"2 3(6;7) 2 = 6 + " 2 3 = 6 + " ; " 2 4 = 6 + (2;2)" 3(2;) + (32;)"2 : 3(6;7) If 4, the matrix A has := b( ; )=2c; double eigevalues besides. We deote these eigevalues by ::: ad assume j j::: j j. Furthermore, ay eigevalue of A with modulus i the half-ope iterval [jj 2 jj) is oe of these double eigevalues i but ot vice versa. Chagig just these double eigevalues i to the double eigevalues i + i results i a matrix A, which represets the same mass as the origial matrix except for the E-mas illustrated i Figure 5, where X i = f i + 2 2i(j +) j cos j= ad f i = < : 3= = if i = i = i 2 i = ::: b=2c : (5.2) are show with the visualizatio of their Gaussia curvature. This curvature is ot a discrete approximatio obtaied from the subdivided cotrol et. We used the piecewise quartic parametrizatio of the surface to compute the Gaussia curvature. The commo cotrol et of both surfaces is give i Figure /2 3/ 2 Figure 6. Visualizatio of the Gaussia curvature of the surface geerated from the et show i Figure 7 by Loop s algorithm (top) ad our modificatio (bottom). 2 Figure 5. The E-mass of the modified Loop algorithm ear the vertices of valece 4 illustrated for =. Note that Loop s mass, see Figure, are obtaied if all s ad " s are zero. Figure 6 shows a example. The surface at the top is geerated usig Loop s algorithm while the oe at the bottom is produced with the above modified mass, where = :3755 ad 2 = ::: =. The surfaces Figure 7. Topview of the cotrol et used for Figure 6. It lies o a parabolic cylider.

5 Remar 5.2 The eigevalues of A with modulus less tha jj 2 eed ot be chaged. However, the mass of the modified algorithms have egative weights, see (5.2). Therefore the eigevalues with modulus < 2 ca be chaged so as to obtai larger egative weights. Remar 5.3 I some cases better looig surfaces are obtaied if Loop s algorithm is gradually modified after each subdivisio iteratio. For example, the sequece of ets N i i = ::: 6 leadig to the surface show i Figure (bottom left) has bee obtaied by Loop s algorithm modified with " =2" 2 = p i=34 whe applied to the et Ni. The adaptive liear combiatio of Loop s ad our scheme produces a surface with a more eve curvature distributio ad without ifiite curvature. I further iteratios we would chose " ad " 2 costat as i step 6. Note that the modified subdivisio matrix satisfies the coditios of Theorem 4. for i Figure. Visualizatio of the Gaussia curvature of the surface geerated from the et show i Figure 9 by Loop s algorithm (top left), our modified scheme (top right) ad a adaptive liear combiatio of Loop s ad our scheme (bottom left). Figure 9. Topview of the cotrol et used for Figure. It lies o a hyperbolic paraboloid. 6. Modificatios of the butterfly algorithm A limitig surface obtaied by the butterfly algorithm is ot differetiable at extraordiary poits, i geeral. For a extraordiary poit of valece 3, this is due to the fact that the associated subdivisio matrix A has a triple subdomiat eigevalue, see [5]. Two of these eigevalues are associated with eigevectors formig a regular ijective characteristic map. As i Sectio 5 we chage the third eigevalue to ; :. For a extraordiary poit of valece the characteristic map of the subdivisio matrix A overlaps itself. Sice the matrices are bloc circulat they have a discrete Fourier trasform, which helps to uderstad what happes: The subdomiat eigevalues for correspod to higher frequecies tha oe. Lucily both eigevectors associated with the largest eigevalue of frequecy oe (it is a double real eigevalue, here deoted by ) represet the cotrol et of a regular ijective surface rig. Thus we chage the eigevalues with modulus i ( jj) to ; : as i Sectio 5 so that becomes the subdomiat eigevalue. Together these chages lead to a modificatio of the butterfly algorithm for! = =32 which is preseted by the same mass except for the mass give i Figure, where the weights i i i are give i Table. Figure 2 shows a example. The surface at the top is geerated usig the butterfly scheme while the oe at the bottom is produced with the above modified mass. Note that the surface at the top has self-itersectios while the surface at the bottom as well as the commo cotrol et of both surfaces, see Figure 3, have o self-itersectios. Remar 6. The surface obtaied by the modified butterfly algorithm does ot iterpolate all vertices of the iitial cotrol et. However, if we use the butterfly algorithm i the first iteratio ad the modificatio i all further iteratios, all vertices of the iitial et are iterpolated.

6 =3 = 6 6 :: 6 :: 6 :: 6 :: 4 :: 4 :: 3 :5 : :625 :4936 :9566 :63594 ;:63 ;:29 : ;:3744 :2 ;:339 :52437 :2942 :95 :625 : : :49542 :969 :6439 : : ;:23677 ;:22 ;:25 :49452 ;:55 ;:2 :69 ;:346 :952 ;:355 ;:24256 :433 ;:94 :26392 ;:3 :9 :63 :29 :7572 ;:22 ;:25 ;:7347 ;:55 ;:2 :69 :33 :952 ;:265 : : : : : ;:7572 :55 :2 ;:69 ;:6993 ;:433 :94 ;:63 ;:29 =9 = :: 5 :: 5 :: 5 :: 7 :: 7 :: 6 :5 : :625 :5 : :625 :47959 :97623 :64752 :4967 :97322 :6433 ;:37 ;:27 :447 ;:2 ;:223 :4666 ;:339 :293 ;:3432 ;:3342 :799 ;:336 :5779 ;:735 :696 :66 ;:3764 :327 ;:42 ;:3 ;:254 ;:9 ;:3 ;:2256 ;:424 ;:37 :69 ;:592 ;:74 :9 ;:259 :9243 ;:22 ;:2976 :237 ;:769 : :29 :9767 :5 :29 :633 ;:372 ;:27 :225 ;:3 ;:227 :6 :974 :9243 ;:22 :5466 :3 ;:94 : :29 :6 :56 : ;:37 ;:424 ;:37 :69 :25 :47 ;:3 ;:372 ;:735 :696 ;:374 ;:666 :442 ;:42 ;:3 ;:5992 ;:6 ;:7 ;:7379 : :293 ;:2 :4 :2349 ;:7 ;:24 ;:2376 :2252 ;:37 ;:27 = =2 +2 E-mas E-mas V-mas Figure. The E- ad V-mass of the modified butterfly algorithm ear a vertex of valece =. :: :: :: :: 2 :: 2 :: 9 :5 : :625 :5 : :625 :47942 :95659 :6636 :452 : :643 ;:235 ;:472 :47393 ;:47 ;:29 :4363 ;:29375 :3762 ;:3566 ;:363 :2354 ;:3459 :5694 ;:3 :9 :5757 ;:9764 :99 ;:74 ;:4 ;:2999 ;:6 ;:2 ;:24 ;:64 ;:32 :93 ;:2 ;:39 :35 ;:623 :3236 ;:267 ;:2349 :5393 ;:56 :6 :323 :2696 :96 :9 :922 ;:74 ;:3762 :36 ;:353 ;:265 :23 :3 :2227 ;:2379 :36 :22553 ;:226 :4 :297 :645 :4 :2 :35 ;:55 ;:996 :29 ;:923 ;:32 :62 ;:395 ;:2 :292 :66 :3345 ;:339 ;:2 ;:25 ;:25 :2 :4 ;:33 :64 :32 ;:93 :56 :62 ;:3 ;:555 ;:3 :9 ;:9967 ;:9774 :24 ;:74 ;:4 ;:645 ;:23 ;:245 ;:23 :55 :996 ;:29 :4 :2799 ;:24 ;:636 ;:375 :97 ;:62 ;:22 Table. The weights of the mass of the modified butterfly algorithm for =3 ad = ::: 2. Figure 2. The surface geerated from the et show i Figure 3 by the butterfly scheme (top) ad our modificatio (bottom).

7 Figure 3. The cotrol et used for Figure Acowledgemets We wish to tha Uwe Klotz who helped us to geerate Figures 6, ad 2. [] H. Prautzsch ad G. Umlauf. A G 2 -subdivisio algorithm. To appear i: Bieri, Bruet, Fari, editors, Proceedigs of the Dagstuhl coferece o geometric modellig 996, Computig suppl., 99. [] U. Reif. A uified approach to subdivisio algorithms ear extraordiary vertices. Computer Aided Geometric Desig, 2:53 74, 995. [2] U. Reif. A Degree Estimate for Subdivisio Surfaces of Higher Regularity. Proceedigs of the America Mathematical Society, 24(7): , 996. [3] M. Sabi. Cubic Recursive Divisio With Bouded Curvature. I P. Lauret, A. L. Méhauté, ad L. Schumaer, editors, Curves ad Surfaces, pages Academic Press, Bosto, 99. [4] G. Umlauf. Verbesserug der Glattheitsordug vo Uterteilugsalgorithme für Fläche beliebiger Topologie. Master s thesis, IBDS, Uiversität Karlsruhe, Apr [5] G. Umlauf. Aalyzig the characteristic map of triagular subdivisio schemes. Submitted, [6] D. Zori, P. Schröder, ad W. Sweldes. Iterpolatory Subdivisio for Meshes with Arbitrary Topology. Computer Graphics (ACM SIGGRAPH 96 Proceedigs), pages 9 92, 996. Refereces [] A. Ball ad D. Storry. A matrix approach to the aalysis of recursively geerated B splie surfaces. Computer Aided Desig, (): , 96. [2] A. Ball ad D. Storry. Coditios for Taget Plae Cotiuity over Recursiveley Geerated B splie Surfaces. ACM Trasactios o Graphics, 7(2):3 2, 9. [3] E. Catmull ad J. Clar. Recursive geerated B splie surfaces o arbitrary topological meshes. Computer Aided Desig, (6):35 355, 97. [4] D. Doo ad M. Sabi. Behaviour of recursive divisio surfaces ear extraordiary poits. Computer Aided Desig, (6):356 36, 97. [5] N. Dy, J. Gregory, ad D. Levi. A Butterfly Subdivisio Scheme for Surface Iterpolatio with Tesio Cotrol. ACM Trasactios o Graphics, 9(2):6 69, 99. [6] N. Dy, D. Levi, ad C. Micchelli. Usig parameters to icrease smoothess of curves ad surfaces geerated by subdivisio. Computer Aided Geometric Desig, 7:29 4, 99. [7] C. Loop. Smooth Subdivisio Surfaces Based o Triagles. Master s thesis, Departmet of Mathematics, Uiversity of Utah, Aug. 97. [] H. Prautzsch. Aalysis of C subdivisio surfaces at extraordiary poits. Preprit 4/9, Faultät für Iformati, Uiversität Karlsruhe, [9] H. Prautzsch ad U. Reif. Necessary Coditios for Subdivisio Surfaces. Soderforschugsbereich 44, Uiversität Stuttgart, Bericht 97/4, 997.