GEORG UMLAUF. Abstract. Tools for the analysis of generalized triangular box spline subdivision schemes are

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1 ANALYZING THE CHARACTERISTIC MAP OF TRIANGULAR SUBDIVISION SCHEMES GEORG UMLAUF Abstract. Tools for the aalysis of geeralized triagular box splie subdivisio schemes are developed. For the rst time the full aalysis of Loop's algorithm ca be carried out with these tools. Key words. Subdivisio, triagular schemes, Loop's algorithm, box splies. AMS subject classicatios. 65D17, 65D07, 68U07 Abbreviated title. Characteristic Map of Triagular Subdivisio Schemes 1. Itroductio. I the last two decades may subdivisio schemes for dieret purposes were devised, see e.g. [, 1, 6, 11, 3, 4, 7]. All these algorithms geerate from a iitial cotrol et a sequece of ets which coverges to a limitig surface. Although suciet coditios for the covergece to a smooth limitig surface were give i [1] ad [9], their rigorous applicatio has bee carried out oly for some of the above metioed schemes by Peters ad Reif [8, 7]. I this paper we will ivestigate the smoothess of the limitig surfaces obtaied by subdivisio algorithms for triagular ets. Accordig to the suciet coditios of [1] ad [9], we have to aalyze the spectral properties of the subdivisio matrix associated with the algorithm ad the so-called characteristic map. Symmetry properties of the algorithms help to simplify this aalysis sigicatly. Subsequetly the aalysis is carried out for Loop's algorithm [6]. The spectral properties of the subdivisio matrix imply some characteristics of the algorithm as already observed by Loop [6]. To prove regularity ad ijectivity of the characteristic map we use its Bezier represetatio as i [13] ad[8]. This leads to a rigorous proof of taget cotiuity of the limitig surface of Loop's algorithm.. Geeralized subdivisio. We presume that all subdivisio algorithms cosidered here are statioary, local, ad liear schemes for tri- or quadrilateral ets. Such a algorithm geerates startig from a iitial arbitrary tri- or quadrilateral et C 0 a sequece of ever er ets fc m g 1 m=0. Thereby oly ite, ae combiatios, represeted by so called masks, are used to compute the poits of the et C m from C m;1 m 1. This makes up for the locality ad liearity of these schemes. Sice we use the same ae combiatios i every step m of the iteratio, the subdivisio algorithm is said to be statioary. The sequece of ets fc m g 1 m=0 geerated by such a algorithm will evetually coverge to a limitig surface s cosistig of iitely may tri- or quadrilateral patches. A example for Loop's algorithm is show i Figure.1. The upper left et is the iitial et C 0. The other ets C 1 ::: C 4 are the result of the rst four iteratios of Loop's algorithm startig from C 0. Suppose that o the regular parts of a et, i.e. parts of the et that cotai oly ordiary vertices of valece 6 or 4 for tri- or quadrilateral ets respectively, stadard Fakultat fur Iformatik, Uiversitat Karlsruhe, D-7618 Karlsruhe, Germay, umlauf@ira.uka.de. Supported by DFG grat PR 565/1-1. 1

2 GEORG UMLAUF C 0 C 1 C C 3 C 4 Fig..1. The iitial triagular et C 0 (top left) ad the ets C 1 ::: C 4 of the rst four iteratio steps of Loop's algorithm. subdivisio rules for symmetric box splies apply. Examples for such stadard subdivisio rules are the subdivisio rules for tesorproduct splies [5] or for quartic box splies over the three-directioal grid. O this coditio the regular parts of a et determie C k -surfaces. Near vertices of valece 6= 6(6= 4) for tri- (quadri-)lateral ets, the so-called extraordiary vertices, special subdivisio rules are used, which dootchage the umber of extraordiary vertices i two cosecutive ets C m;1 ad C m, m. Sice the subdivisio masks of statioary, local schemes have xed ite size, we ca restrict the aalysis to ets C 0 with a sigle extraordiary vertex surrouded by r rigs of ordiary vertices. A example is illustrated i Figure. for r =3. Fig... A iitial et C 0 with a extraordiary vertex of valece 5 (marked by) surrouded by r =3rigs of ordiary vertices. The particular choice of r depeds o the subdivisio algorithm. It must be

3 CHARACTERISTIC MAP OF TRIANGULAR SUBDIVISION SCHEMES 3 such that the regular parts of C 0 dee at least oe complete surface rig. Loop's algorithm requires for example r = 3. If we deote by s m the surface that correspods to the regular parts of C m, the the limitig surface is give by s = [s m. Obviously s m;1 is part of s m for m 1. So takig s m;1 away froms m we obtai a surface S rig r m which is added to s m;1 i the mth iteratio step. This yields s = s 0 [ r m1 m. At a extraordiary vertex of valece the surface rigs r m ca be parametrized over a commo domai Z i terms of a subet D m C m ad certai fuctios N k.ifd 0 m ::: dk m deote the vertices of D m,wehave where is either (u v j) 7! r j m(u v) = i case of trilateral ets or r m : Z! IR 3 KX k=0 d k m N k (u v j) =:N(u v j) d m 4 = f(u v)ju v 0 ad 1 u + v g = f(u v)ju v 0 ad 1 maxfu vgg i case of quadrilateral ets, see Figure.3. p p e 4 e e 1 e 1 Fig..3. The domais 4 (left) ad (right). Note that all ets D m have equally may vertices. Hece the statioary, local, ad liear subdivisio algorithm ca be described by a square subdivisio matrix A, i.e. d m = Ad m;1 : 3. The subdivisio matrix ad the characteristic map. Let 0 ::: K be the eigevalues of A listed with all their algebraic multiplicities ad ordered by their modulus j 0 jj 1 j:::j K j ad deote by v 0 ::: v K the correspodig geeralized eigevectors. If j 0 j > j 1 j = j j > j 3 j, the two dimesioal surface that is deed by the et [v 1 v ] x(u v j) :=N(u v j)[v 1 v ]: Z! IR

4 4 GEORG UMLAUF x 1 x 0 x ;1 Fig The characteristic map of Loop's algorithm for =7. is called the characteristic map of the subdivisio scheme [1]. Note that x ca be regarded as cosistig of segmets x j (u v) :=x(u v j). A example for the characteristic map of Loop's algorithm is show i Figure 3.1. The crucial theorem for the aalysis of subdivisio algorithms ca ow be stated i terms of the subdomiat eigevalue := = 3 ad x: Theorem 3.1. Let be areal eigevalue with geometric multiplicity. If the characteristic map x is regular ad ijective ad 0 =1> jj > j 3 j the the limitig surface isac 1 -maifold for almost all iitial cotrol ets C 0. Proofs of this theorem ca be foud i [1] or i a more geeral settig i [9]. I the sequel we apply Theorem 3.1 to subdivisio schemes with two additioal properties. 1. A subdivisio algorithm is said to be symmetric, if it is ivariat uder shifts ad reectios of the labellig of d m. This meas if permutatio matrices S ad R characterized by N(u v j +1)d m = N(u v j)sd m ad exist, the A commutes with S ad R: N(v u ;j)d m = N(u v j)rd m AS = SA ad AR = RA: Note that S ad R exist especially for subdivisio algorithms based o box splies with regular hexagoal or square support.. A subdivisio algorithm is said to have a ormalized characteristic map, if x 0 (p) =(p 0) with p >0adp = e 1 + e or p =e 1 +e i case of tri- or quadrilateral ets, respectively, see Figure.3. The rst property implies that the subdivisio matrix A has a block-circulat structure with square blocks A j j =0 ::: ; 1. Thus A is uitary similar to a

5 CHARACTERISTIC MAP OF TRIANGULAR SUBDIVISION SCHEMES 5 block-diagoal matrix A. b The diagoal blocks of A b result from Aj by the discrete Fourier trasform: ;1 X ba j = k=0! ;jk A k for j =0 ::: ; 1 where! =exp(i=) deotes a -th root of uity. This meas, if bv is a eigevector of some block b Aj correspodig to the eigevalue, the is also a eigevalue of A with eigevector (3.1) v =[! bv! 0 bv :::! j j(;1) bv]: If deotes the complex-cojugate, the blocks of b A satisfy b Aj = b A ;j for j = 1 ::: b=c. Hece there are always two liear idepedet, real eigevectors v 1 = <(v) ad v = =(v) correspodig to the real subdomiat eigevalue. From this a rst ecessary coditio for the subdomiat eigevalue ca easily be deduced [8]: Lemma 3.. The characteristic map of a symmetric subdivisio scheme i ot ijective, if the subdomiat eigevalue is from a block b Aj for j 6= 1 ; 1. Equatio (3.1) shows also that ormalizatio of a ijective characteristic map ca always be achieved by a appropriate scalig of bv. 4. Suciet coditios for regularity ad ijectivity. Throughout this chapter we will assume that the subdomiat eigevalue is a real eigevalue from the blocks b A1 ad b A;1. The two properties of the last chapter imply that the characteristic map x is symmetric uder rotatios ad reectios for subdivisio schemes for tri- or quadrilateral ets ([8]). They allow us to restrict the aalysis of the characteristic map to its sigle segmet x 0 : Theorem 4.1. Let x 0 =[x y] ad deote by x 0 v := [x v y v ] the partial derivatives of x 0 with respect to v. If the ormalized characteristic map x of a symmetric subdivisio scheme for quadrilateral ets satises x 0 v(u v) > 0 for all (u v) compoetwise, the the characteristic map is regular ad ijective. For a proof of this theorem see [8]. The proof also applies to ay map z : Z! IR that shares the above symmetry properties of the map x. Theorem 4.1 ca be trasfered to triagular ets by the observatio that ay triagular et as i Figure. ca be viewed as a quadrilateral et with diagoal edges. This is show i Figure 4.1 for the domais 4 ad. 4 4 Fig Covertig a triagular to a quadrilateral et.

6 6 GEORG UMLAUF I case of triagular ets the domai of the characteristic map y cosists of plae copies of 4.Further we have 4 [ 4, as illustrated i Figure 4.1. Dee the map z as z : Z! IR y (u v j) 7! j (u v) if (u v) 4 y j (u v) if (u v) 4 : Thus z is "covered" by y ad y ad adopts its symmetry properties. Hece Theorem 4.1 is valid for the map z ad ca be applied to subdivisio schemes for triagular ets, if x is replaced by y ad by 4 : Theorem 4.. If for the ormalized characteristic map y of a symmetric subdivisio scheme for triagular ets the segmet y 0 satises y 0 v(u v) > 0 for all (u v) 4 compoetwise, the the map z is regular ad ijective. I practice we will use the Bezier represetatio of y 0 to apply Theorem 4.. Hece the proof of the positivity ofy 0 v for all (u v) 4 reduces to the proof of the positivity of its Bezier poits. Corollary 4.3. If all Bezier poits of y 0 v are positive, the the ormalized characteristic map y of a symmetric subdivisio scheme for triagular ets is regular ad ijective. 5. Loop's algorithm. Loop's algorithm is a geeralizatio of the subdivisio scheme for quartic box-splies over a regular triagular grid. The masks are give i Figure 5.1, where the parameter ca be chose arbitrarily from the iterval ; ; cos(=)=4 (3 + cos(=))=4 : (5.1) 3=8 1=8 1=8 3=8 Fig The masks of Loop's algorithm. The limitig surface geerated by this algorithm is a piecewise quartic C -surface except at its extraordiary poits. Here the surface is cojectured to be taget cotiuous, see [6]. Obviously, Loop's algorithm is a symmetric scheme. The form of its subdivisio matrix A depeds o the labellig of the vertices i the cotrol et D m.ifwelabel them segmet after segmet couterclockwise as i Figure 5. the subdivisio matrix A has a block-circulat structure: A = 6 4 A 0 A 1 A ;1 A ;1 A 0 A ; IR77 : A 1 A ;1 A 0 3

7 CHARACTERISTIC MAP OF TRIANGULAR SUBDIVISION SCHEMES 7 d 1 d 7 i = i = ::: d i d 8 i d 4 i d 9 i d 7 i d 3 i d 6 i d 7;1 i d 7 i d 5 i Fig. 5.. The labellig of the vertices of the cotrol et D m for =5. Now we ca use the discrete Fourier trasform as i Chapter 3 to yield a uitary similar block-diagoal matrix b A with blocks ba 0 = ba j = ; 3=8 5=8 1=16 3=4 1=16 1=8 1=8 3=4 0 1=8 0 3=8 3=8 1= = 1=8 3= = 1=8 3= =8+c j =4 0 5=8+c j =8 1=16 1=16 +!j =16 ad 0 3=8+3! ;j =8 0 1=8 0 3=8 3=8 1=8+! j = =8+! ;j 0 1=8+3! ;j =8 1=8 3= =8 3= =8!;j for j =1 ::: ; 1ad c j + isj subdivisio matrix A as follows: 1, := ; 3=8, j := 3=8+c j =4 for j =1 ::: ; 1, =!j. From this we get the eigevalues of the 1=8 ad1=16 each -fold ad 0 which occurs (4 ; 1)-fold. Note that j = ;j for j =1 ::: b=c ad 1 > j for j = ::: b=c. Furthermore b A1 = b A ;1,sothat 1 has geometric multiplicity. Therefore 1 is the double subdomiat eigevalue of A, ifj j < 1. This last iequality yields the iterval (5.1) for give by [6]. This veries the coditios of Theorem 3.1 for the subdomiat eigevalue of the subdivisio matrix. What remais is the aalysis of the characteristic map. Remark 5.1. The spectral aalysis of A ca also be used to modify the subdivisio

8 8 GEORG UMLAUF algorithm so as to geerate smoother limitig surfaces [10]. 6. The characteristic map of Loop's algorithm. Accordig to Corollary 4.3 we oly eed to show positivity of the Bezier poits of y 0 v to proof regularity ad ijectivity of the characteristic map y of Loop's algorithm. Some calculatios usig a computer algebra system yield the Bezier poits of y 0 v as give i Figure 6.1. Except for positive costats the deomiators are give by D 1 =5+4c 1 D =54+36c 1 D 3 =19+c1 +4c : Obviously D 1 D D3 are positive for arbitrary 3 sice c 1 ;1=. The umerators i Figure 6.1 are of the form E c = a 0 + a 1 c 1 + a c + a 3 c 3 a 0 a 1 a a 3 Z or E s = b 1 s 1 + b s + b 3 s 3 b 1 b b 3 Z: Sice a 0 a 1 > 0, the umerators E c are positive if a 0 >a 1 =+ja j + ja 3 j: This last coditio is fullled by all E c. The positivity of the other umerators Es for 3 ca be show i a similar fashio. This completes the proof for Lemma 6.1. Loop's algorithm geerates C 1 -maifolds for almost all iitial triagular ets C 0. Remark 6.. This proof applies also to the subdivisio schemes proposed i [10]. Thus these schemes geerate curvature cotiuous surfaces with at spots at the extraordiary poits for almost all iitial triagular ets C 0. REFERENCES [1] E. Catmull ad J. Clark, Recursive geerated b-splie surfaces o arbitrary topological meshes, Computer Aided-Desig, 10 (1978), pp. 350{355. [] D. Doo ad M. Sabi, Behaviour of recursive divisio surfaces ear extraordiary poits, Computer Aided-Desig, 10 (1978), pp. 356{360. [3] N. Dy, J. Gregory, ad D. Levi, A buttery subdivisio scheme for surface iterpolatio with tesio cotrol, ACM Trasactios o Graphics, 9 (1990), pp. 160{169. [4] L. Kobbelt, Iterative Erzeugug glatter Iterpolate, PhD thesis, IBDS, Uiversitat Karlsruhe, Verlag Shaker, Aache. [5] J. Lae ad R. Riesefeld, A Theoretical Developmet for the Computer Geeratio ad Display of Piecewise Polyomial Surfaces, IEEE Trasactios o Patter Aalysis ad Machie Itelligece, (1980), pp. 35{46. [6] C. Loop, Smooth Subdivisio Surfaces Based o Triagles, master's thesis, Departmet of Mathematics, Uiversity of Utah, Aug [7] J. Peters ad U. Reif, The simplest subdivisio scheme for smoothig polyhedra, ACM Trasactios o Graphics, 16 (1997), pp. 40{431. [8], Aalysis of algorithms geeralizig b-splie subdivisio, SIAM Joural o Numerical Aalysis, 35 (1998), pp. 78{748. [9] H. Prautzsch, Aalysis of C k -subdivisio surfaces at extraordiary poits, preprit 98/4, Fakultat fur Iformatik, Uiversitat Karlsruhe, [10] H. Prautzsch ad G. Umlauf, Improved triagular subdivisio schemes, i Computer Graphics Iteratioal 1998, F.-E. Wolter ad N. Patrikalakis, eds., Haover,.-6. Jui 1998, IEEE Computer Society, pp. 66{63. [11] R. Qu, Recursive subdivisio algorithms for curve ad surface desig, PhD thesis, Departmet of Mathematics ad Statistics, Burel Uiversity, Uxbridge, Middlesex, Eglad, Aug

9 CHARACTERISTIC MAP OF TRIANGULAR SUBDIVISION SCHEMES 9 " 1 ;c ;4c c 3 9D " 1 +4c ;4c c 3 9D " 1 ;7c ;c c 1 18D " 1 ;c ;c c 1 18D " 1 ;c 84+80c 3 9D " 1 +c ;c c 3 6D " 1 ;6c ;4c c 3 18D " 1 ;5c 44+33c 1 18D " 1 ;6c ;8c c 3 18D " 1 ;6c ;c c 3 6D " 1 +c 44+39c 1 18D " 1 ;7c 46+7c 1 18D " 1 ;18c ;10c c 3 18D " 1 ;30c ;8c c 3 18D " 1 +c 44+39c 1 18D " 1 ;11c 47+1c 1 18D " 1 ;18c ;7c c 3 9D " 1 ;17c ;4c c 1 18D " 1 ;1c ;c c 1 18D " 1 ;8c ;c c 3 3D " 1 ;30c ;10c c 3 9D " 1 ;34c ;8c c 3 9D 1 +6s ;4s 3 78s 3 9D 1 +1s ;4s 3 7s 3 9D 1 ;3s ;s 3 45s 1 18D 1 +3s ;s 3 39s 1 18D 1 +1s ;s 3 7s 3 9D 1 +18s ;s 3 66s 3 9D 1 +s 13s 1 6D 1 +3s 11s 1 6D 1 +6s s 3 3D 1 +4s +s3 60s 3 9D 1 +3s 11s 1 6D 1 +s 5s s +s3 60s 3 9D 1 +4s 0s D 1 +5s 9s 1 6D 1 +1s +4s3 1s 1 18D 1 +4s 0s D 1 +36s +7s3 48s 3 9D 1 +1s +s3 1s 1 18D 1 +1s +s3 16s 3 3D 1 +4s +10s3 4s 3 9D 1 +4s +8s3 4s 3 9D Fig The Bezier poits of y 0 v.

10 10 GEORG UMLAUF [1] U. Reif, A uied approach to subdivisio algorithms ear extraordiary vertices, Computer Aided Geometric Desig, 1 (1995), pp. 153{174. [13] J. Schweitzer, Aalysis ad Applicatio of Subdivisio Surfaces, PhD thesis, Departmet of Computer Sciece ad Egieerig, Uiversity of Washigto, Seattle, Washigto 98195, Aug Techical Report UW-CSE