A study on the mask of interpolatory symmetric subdivision schemes

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1 Appled Mathematcs and Computaton 87 (007) A study on the mask of nterpolatory symmetrc subdvson schemes Kwan Pyo Ko a, Byung-Gook Lee a, Gang Joon Yoon b, * a Dvson of Computer and Informaton Engneerng, Dongseo Unversty, Busan 67-76, South Korea b School of Mathematcs, Korea Insttute for Advanced Study, Cheongnyangn dong, Dongdaemun-gu, Seoul 30-7, South Korea Abstract In the work, we rebuld the masks of well-known nterpolatory symmetrc subdvson schemes-bnary n-pont nterpolatory schemes, the ternary 4-pont nterpolatory scheme usng only the symmetry and the necessary condton for smoothness and the butterfly scheme, and the modfed butterfly scheme usng the factorzaton property. Ó 006 Elsever Inc. All rghts reserved. Keywords: Mask; Interpolatory symmetrc subdvson scheme; Ternary scheme; The butterfly scheme. Introducton Subdvson schemes are a very effcent tool for the fast constructon of smooth curves and surfaces from a set of control ponts through teratve refnements. In recent years, subdvson schemes became one of the most popular methods of creatng curves and surfaces n computer aded geometrc desgn (CAGD) and n the anmaton ndustry. Now, they have become a subect of study n ther own rght wth a varety of applcaton to computer graphcs, geometrc modelng and wavelets. Furthermore, schemes have been appled to solve problems n flud flow. Ther popularty s due to the facts that subdvson algorthms are smple to apprehend, sutable for computer applcatons, easy to mplement. Also, the subect of subdvson schemes has an nterdscplnary relaton to wavelet, whch has brought on recprocal mprovement n both the areas. Each subdvson scheme s assocated wth a mask a ¼fa a R : a Z s g, where s = n the curve case and s = n the surface case. The (statonary) subdvson scheme s a process whch recursvely defnes a sequence of control ponts ¼f a : a Zs g by a rule of the form wth a mask a ¼fa g Z s a ¼ a a Mb b ; k f0; ; ;...g; bz s whch s denoted formally by + = S. Then a pont of + s defned by a fnte affne combnaton of ponts n. Here M s an s s nteger matrx such that lm n! M n = 0. The matrx M s called a dlaton * Correspondng author. E-mal addresses: kpko@dongseo.ac.kr (K.P. Ko), lbg@dongseo.ac.kr (B.-G. Lee), yk@kas.re.kr (G.J. Yoon) /$ - see front matter Ó 006 Elsever Inc. All rghts reserved. do:0.06/.amc

2 60 K.P. Ko et al. / Appled Mathematcs and Computaton 87 (007) matrx. Bnary (or dyadc) and ternary subdvson schemes are schemes wth the matrces M =I and M =3I, respectvely, for the s s dentty matrx I. Among subdvson schemes, an nterpolatory subdvson s more ntutve because t preserves the data obtaned at the former stage and refnes new data by nsertng values correspondng to ntermedate ponts. The nserted values are determned by lnear combnatons of values at the neghborng ponts. An nterpolatory symmetrc bnary scheme s one of the most popular subdvson schemes n ther applcatons. Ths results n ts smplcty to use and the tendency that customers feel comfortable for symmetrc obects. Despte ts smplcty, an nterpolatory bnary subdvson scheme has the drawback that n order to create smoother curves or surfaces, t s necessary to enlarge the support of the mask. It s well-known, n general, that the creaton of hghly smooth curves or surfaces n a subdvson scheme and the shortness of the support sze of ts mask are two mutually conflctng requrements, n spte that desgners n CAGD requre subdvson schemes to have ther masks wth a possbly smaller support and to create good smooth curves or surfaces. For example, t was proved n [8] that there s no C nterpolatory bnary subdvson scheme wth a mask supported on [ 3,3] s. There are two approaches to acheve a desred mask: one s to enlarge the sze of the support so that we may allow free parameters n the mask; the other s to fx the sze and fnd specfc values of the mask. There s no unque method of obtanng a mask. Deslaurers and Dubuc [] obtaned the mask of n-pont nterpolatory subdvson rule by polynomal reproducng property. Dyn [3] commented that we can get the mask of 4-pont and 6-pont schemes wth a one parameter by takng a convex combnaton of the two Deslaurers and Dubuc schemes. In the work, we try to fnd smoother possble masks wth the sze fxed by applyng repeatedly some smoothness condtons. Then, we rebuld the masks of nterpolatory symmetrc subdvson schemesbnary n-pont nterpolatory schemes, ternary 4-pont nterpolatory scheme-n Sectons 3 and 4. In Secton 5, we study the butterfly scheme and we observe that n ts stencl, the sums of even masks and odd masks along every vertcal or horzontal lne are the same, whch nduces a factorzaton of the correspondng Laurent polynomal (see Theorem 5). By applyng the observaton, we show that from the same structure of the stencl used n the butterfly scheme, we can obtan the mask of the scheme and the modfed butterfly schemes, as well.. Prelmnares Throughout the work, we consder schemes wth a mask of fnte support. That s, the set f Z s : a 6¼ 0g s fnte. Ths property s useful n the practcal mplementaton because changes n a control pont affect only ts local neghborhood. In ths secton, we consder a subdvson scheme S wth a mask a and a dlaton matrx M. We denote by / 0 the tensor product of the symmetrc hat functon n R s defned by / 0 ðxþ :¼ Ys B ðx Þ for ðx ;...; x s ÞR s ; ðþ ¼ where B s the B-splne of degree (order ) gven as 8 >< þ x; for x ½ ; 0Þ; B ðxþ ¼ x; for x ½0; Š; >: 0; for x R n½ ; Š: The subdvson operator S assocated wth a mask a and a dlaton matrx M s the lnear operator on the space ðz s Þ of all sequences on Z s defned by SvðaÞ :¼ bz s a a Mb vðbþ; v ðz s Þ: Defnton. A subdvson scheme S wth a mask a and a dlaton matrx M converges unformly f for every sequence f 0 ¼ðf 0 a Þ az s wth compact support, there exsts a functon f CðRs Þ such that

3 lm S n f 0 n! a / 0ðM n aþ f ¼ 0; az s where / 0 s the unt hat functon gven n (). And for some sequence, the above functon f s not dentcally zero. The functon f s denoted by S f 0. A unformly subdvson scheme s sad to be C m f for any ntal data the lmt functon has contnuous dervatves up to order m. Let E be a complete set of representatves of the dstnct cosets of Z s =MZ s. Then Z s s the dsont unon of c þ MZ s, c E. Also, t s not dffcult to see that the set fm n a : a Z s, n Ng s dense n R s. Theorem. Let S be a subdvson scheme wth a mask a and a dlaton matrx M. If S converges unformly, then for every c E, we have a c Ma ¼ : ðþ az s Proof. Let f 0 be an ntal data such that S f 0 5 0, and let c E be arbtrarly fxed. From () and Defnton, we can see that lm n! sup az s S n f 0 ðaþ f ðm n aþ ¼ 0: K.P. Ko et al. / Appled Mathematcs and Computaton 87 (007) By the contnuty of f, there are a Z s and m N such that f(m m a) 5 0. For n P max(m,n), we have that S n f 0 ðm n m a þ cþ ¼ bz s a M n m aþc MbS n f 0 ðbþ ¼ bz s a Mbþc S n f 0 ðm n m a bþ: Wth the equaton above, we nduce that f ðm m a þ M n cþ bz s a Mbþc f ðm m a M nþ bþ ¼ f ðm m a þ M n cþ S n f 0 ðm n m a þ cþþ bz s a Mbþc ðs n f 0 ðm n m a bþ f ðm m a M nþ bþþ: On the other hand, snce the support of the mask a s fnte, we can exchange the lmt and the summaton as we take n!. Hence, by the unform convergence of S and the contnuty of f, we obtan that f ðm m aþ¼ bz s a cþmb f ðm m aþ: Snce f(m m a) 5 0, we have, as a consequence, that a cþmb ¼ ; c E: bz s For the case when M s two tmes the s s dentty matrx, Theorem was proved by Cavaretta et al. [], and Dyn [5]. Han and Ja [9] showed that the condton n () wth P az sa a ¼detðMÞ s necessary for the subdvson scheme to converge n the L p -norm. 3. Bnary n-pont nterpolatory symmetrc subdvson scheme A bnary unvarate subdvson scheme s defned n terms of a mask consstng of a sequence of coeffcents a ¼fa : Zg, that s, a unvarate scheme wth a dlaton matrx M =. The scheme s gven by ¼ Z a ; Z:

4 6 K.P. Ko et al. / Appled Mathematcs and Computaton 87 (007) For each scheme S wth a mask a, we defne the symbol aðzþ ¼ a a z a : az The correspondng symbols, called Laurent polynomals, play a effcent role n analyzng the smoothness of a subdvson scheme. Interpolatory subdvson schemes retan the ponts of stage k as a subset of the ponts of stage k +. Thus, the general form of an nterpolatory subdvson scheme s ; þ ¼ a þ : Z From Theorem, we can see that a ¼ a þ ¼ ð3þ and the Laurent polynomal of a convergent subdvson scheme satsfes að Þ ¼0 and aðþ ¼: ð4þ Ths condton guarantees the exstence of a related subdvson scheme for the dvded dfferences of the orgnal control ponts and the exstence of assocated Laurent polynomal a (z) whch can be defned as follows: a ðzþ ¼ z ð þ zþ aðzþ: ð5þ The subdvson S wth symbol a (z) s related to S wth a(z) by the followng theorem. Theorem [5]. Let S denote a subdvson scheme wth symbol a(z) satsfyng (3). Then there exst a subdvson scheme S wth the property d ¼ S d ; where =S k f 0 and d ¼fðd Þ ¼ k ðfþ k f k Þ : Zg. Furthermore, S s a unformly convergent subdvson scheme f and only f S converges unformly to the zero functon for all ntal data f 0 n the sense that k lm k! S f 0 ¼ 0: ð6þ A scheme S satsfyng (6) for all ntal data f 0 s sad to be contractve. ByTheorem, the check of the convergence of S s equvalent to checkng whether S s contractve, whch s equvalent to checkng whether there exsts an nteger L > 0 such that the operator of L teratons of S satsfes k S L k <. Here, how to compute the norm os L k? From the refnement rule of S (two refnement rules) ¼ a ; we have k k 6! a max k k and we can calculate the norm of S: ( ) ksk ¼ max a ; a þ : We defne generatng functon of control pont as F k ðzþ ¼ z :

5 Snce the coeffcent of z n F k+ (z) s and the coeffcent of z n a(z)f k (z )s P a, we have F kþ ðzþ ¼aðzÞF k ðz Þ: Let a ½LŠ ðzþ ¼ Q L ¼0 aðz Þ¼ P a½lš z, F kþl ðzþ ¼ P f kþl z and well-known fact F kþl ðzþ ¼a ½LŠ ðzþf k ðz L Þ, we have the L refnement rules þl ¼ and norm of S L : ( ) ks L k ¼ max a ½LŠ L L : a ½LŠ L K.P. Ko et al. / Appled Mathematcs and Computaton 87 (007) Theorem 3 [5]. Let aðzþ ¼ ðþzþm bðzþ. If the subdvson S m b correspondng to b(z) s convergent, then S a f 0 C m ðrþ for any ntal data f 0. Therefore, we get the fact that a( ) = 0 s the necessary condton for the convergence of a subdvson scheme and a m ( ) = 0 (where we defne a m ðzþ ¼ z aðzþ; a þz 0ðzÞ ¼aðzÞÞ s the necessary condton for C m - smoothness. Now, we try to fnd the masks of bnary nterpolatory symmetrc subdvson schemes usng only the smoothness condtons pont nterpolatory subdvson scheme The nserton rule of a 4-pont nterpolatory symmetrc subdvson scheme s ; þ ¼ a 3 þ a þ a þ þ a 3 þ : The Laurent polynomal of ths scheme s gven as aðzþ ¼a 3 z 3 þ a z þ þ a z þ a 3 z 3 : Here, a = a and a 3 = a 3 by the symmetrc subdvson scheme condton. And n order for the scheme to be C 0 ðrþ, we have necessarly a( ) = 0, that s, a þ a 3 ¼. In ths case, we obtan the Laurent polynomal of S as a ðzþ ¼ z þ z aðzþ ¼a 3z a 3 z þ þ z a 3z þ a 3 z 3 : Applyng the necessary condton for C ðrþ, we have a ( ) = 0. Set a 3 = w, we fnd the mask of the Dyn 4- pont nterpolatory subdvson scheme: w; þ w; þ w; w : Dyn [5] found a range of the parameter w for smoothness that ths scheme generates C -functons for 0<w < / pont nterpolatory subdvson scheme The general form of 6-pont nterploatory subdvson scheme s gven by ; þ ¼ a 5 þ a 3 þ a þ a þ þ a 3 þ þ a 5 þ3 : The correspondng Laurent polynomal of ths scheme s aðzþ ¼a 5 z 5 þ a 3 z 3 þ a z þ þ a z þ a 3 z 3 þ a 5 z 5 :

6 64 K.P. Ko et al. / Appled Mathematcs and Computaton 87 (007) The symmetrc condton nduces that a = a, a 3 = a 3, a 5 = a 5, and the Laurent polynomal can be wrtten by aðzþ ¼a 5 z 5 þ a 3 z 3 þ a z þ þ a z þ a 3 z 3 þ a 5 z 5 : In order for the scheme to converge unformly, we need the condton a( ) = 0, whch nduces the relaton a þ a 3 þ a 5 ¼. The Laurent polynomal of S s a ðzþ ¼ zaðzþ þ z ¼ a 5 z 4 a 5 z 3 þða 5 þ a 3 Þz ða 5 þ a 3 Þz þ þ z ða 3 þ a 5 Þz þða 3 þ a 5 Þz 3 a 5 z 4 þ a 5 z 5 : To generate C -functons, the Laurent polynomal must satsfy a ( ) = 0. It s easly to see that ths s always true n ths case. And the Laurent polynomal of S becomes a ðzþ ¼ za ðzþ þ z ¼ a 5z 3 a 5 z þð3a 5 þ a 3 Þz ð4a 5 þ a 3 Þþ þ 4a 5 þ a 3 z ð4a 5 þ a 3 Þz þð3a 5 þ a 3 Þz 3 a 5 z 4 þ a 5 z 5 : The necessary condton for C -smoothness mples a ( ) = 0,.e., 4a 5 þ 8a 3 þ ¼ 0. Set a 5 = w, then a 3 ¼ 3w, a 6 ¼ 9 þ w, and we have found the mask of 6-pont nterpolatory subdvson scheme: 6 ; þ ¼ wð þ þ3 Þ 6 þ 3w ð þ þ Þþ 9 6 þ w þ þ : We can easly obtan the mask of 4-pont, 6-pont, 8-pont and 0-pont nterpolatory symmetrc subdvson schemes (ISSS) by usng the same process: 4-pont scheme: ½a ; a 3 Š¼ w þ ; w. 6-pont scheme: ½a ; a 3 ; a 5 Š¼ w 3 þ 9 ; 3w 6 3 ; w pont scheme: ½a ;...; a 7 Š¼ 5w 4 þ 75 ; 9w 8 4 þ 5 ; 5w w pont scheme: ½a ;...; a 9 Š¼ 4w 5 þ 5 ; 8w ; 0w þ 49 ; 7w ; w From these masks, Ko et al. [] found out a general formula for the mask of (n + 4)-pont ISSS wth two parameters whch reproduces all polynomals of degree 6 n + and some relatons between the mask of the (n + 4)-pont ISSS and the (n + )-pont Deslaurers and Dubuc schemes. 4. Ternary nterpolatory symmetrc subdvson scheme A ternary subdvson scheme s a unvarate one wth a dlaton matrx M = 3. The scheme s gven by ¼ Z a 3 ; Z: ð7þ And the general rule of a ternary nterpolatory subdvson scheme s 3 ; 3þ ¼ a þ3 ; Z 3þ ¼ a þ3 : Z

7 K.P. Ko et al. / Appled Mathematcs and Computaton 87 (007) Theorem shows that the mask fa g Z of a convergent ternary subdvson scheme S satsfes a 3 ¼ a 3þ ¼ a 3þ ¼ : The Laurent polynomal of a convergent subdvson scheme satsfes aðe p=3 Þ¼aðe 4p=3 Þ¼0 and aðþ ¼3 and there exsts the Laurent polynomal a (z) such that 3z a ðzþ ¼ ð þ z þ z Þ aðzþ: The subdvson S wth symbol a (z) s related to S wth symbol a(z) by the followng theorem. Theorem 4 [0]. Let S denote a subdvson scheme wth symbol a(z) satsfyng (8). Then there exsts a subdvson scheme S wth the property d ¼ S d ; where =S k f 0 and d ¼fðd Þ ¼ 3 k ðfþ k f k Þ : Zg. And S s a unformly convergent subdvson scheme f and only f S 3 converges unformly to the zero functon for all ntal data f 0 k lm k > 3 S f 0 ¼ 0: Furthermore, S generates C m -lmt functons provded that the subdvson scheme S generates C m -lmt functons for some nteger m P. ð8þ 4.. Ternary 4-pont nterpolatory subdvson scheme We present a ternary 4-pont nterplatory subdvson scheme wth three subdvson rules: 3 ; 3þ ¼ a 4 þ a þ a þ þ a 5 þ ; 3þ ¼ a 5 þ a þ a þ þ a 4 þ : The Laurent polynomal of ths scheme s aðzþ ¼a 5 z 5 þ a 4 z 4 þ a z þ a z þ þ a z þ a z þ a 4 z 4 þ a 5 z 5 : Wth the symmetry condton, a(z) can be wrtten by aðzþ ¼a 5 z 5 þ a 4 z 4 þ a z þ a z þ þ a z þ a z þ a 4 z 4 þ a 5 z 5 : To generate C 0 -functons, we requre that a() = 3. And t mples a + a + a 4 + a 5 =. The Laurent polynomal of S 3 s 3 a ðzþ ¼ z aðzþ þ z þ z ¼ a 5z 3 þða 4 a 5 Þz a 4 z þða 5 þ a Þþð a 5 a Þz þða þ a 5 Þz a 4 z 3 þða 4 a 5 Þz 4 þ a 5 z 5 : From the necessary condton for C -smoothness, the mask correspondng to a (z) satsfes the relaton as n (8),.e., 3a 3a 4 +6a 5 =. And the Laurent polynomal of S 3 s 3 a ðzþ ¼ z a ðzþ þ z þ z ¼ 3a 5z þ 3ða 4 a 5 Þþ3ða 5 a 4 Þz þ 3 3 þ a 4 z þ 3ða 5 a 4 Þz 3 þ 3ða 4 a 5 Þz 4 þ 3a 5 z 5 : To generate C -functons, we requre the mask of S to satsfy the condton (8),.e., a 4 þ a 5 ¼. Set 9 a 5 ¼ þ l, we can fnd the mask of ternary 4-pont nterpolatory subdvson scheme: 8 6

8 66 K.P. Ko et al. / Appled Mathematcs and Computaton 87 (007) ; 3þ ¼ 8 þ 6 l 3þ ¼ 8 6 l þ 3 8 þ l þ 7 8 l þ 8 6 l þ 7 8 l þ 3 8 þ l þ 8 þ 6 l Hassan et al. [0] showed that the scheme s C for 5 < l < Bvarate schemes on regular meshes In ths secton, we consder cases when s = and M s two tmes the dentty matrx. In ths case, we may assume that the complete set E of representatves of the dstnct cosets of Z =Z conssts of the vectors (0,0), (,0), (0, ), and (,). For a quad-mesh, consder the refnement rule for a set of pont f R 3, Z þ ; ¼ Z a ; Z : ð9þ þ : In the bvarate case, there are 4 rules (even even, even odd, odd even and odd-odd) dependng on the party of each component of the vector Z. ð ; Þ ¼ ; a ð ; Þ ð ; Þ ; ðþ ; Þ ¼ ; a ðþ ; Þ ð ; Þ ; ð ;þ Þ ¼ ; a ð ;þ Þ ð ; Þ ; ðþ ;þ Þ ¼ ; a ðþ ;þ Þ ð ; Þ : The easest way to extend unvarate to bvarate schemes s to consder tensor-product schemes. For example, usng the mask of Chakn s scheme, we can get the mask of the bvarate bquadratc scheme a ð;þ : ; a ðþ;þ : ; a ð;þþ : ; a ðþ;þþ : : We can also obtan ths mask set by convoluton of sngle quadratc B-splne mask ½ 3 3 Š. 4 4 Š 4 Š; 4 Š 4 3Š; 4 3Š 4 Š; 4 3Š 4 3Š: Doo Sabn presented an algorthm whch generalze the bquadratc B-splne subdvson rule to nclude arbtrary topology. Smlarly, we can get the mask of b-cubc refnement rule through the tensor product of sngle cubc B-splne mask Š a ð;þ : ; a ðþ;þ : ; a ð;þþ : ; a ðþ;þþ : :

9 K.P. Ko et al. / Appled Mathematcs and Computaton 87 (007) Let us consder the topology of a regular trangulaton. We regard the subdvson scheme as operatng on the three-drectonal grd, snce the three-drectonal grd can be regarded also as Z. Let aðzþ ¼aðz ; z Þ¼ P ; a z z be the symbol of a bvarate subdvson scheme S whch s defned on regular trangulatons. From Theorem the mask of a bvarate convergent subdvson scheme S satsfes a a b ¼ ; a fð0; 0Þ; ð0; Þ; ð; 0Þ; ð; Þg: ð0þ bz It follows that the symbol of a bvarate convergent subdvson scheme satsfes: að ; Þ ¼að; Þ ¼að ; Þ ¼0 and að; Þ ¼4: ðþ In contrast to the unvarate case, the necessary condton (0) and the derved condton () on a(z) do not mply a factorzaton of the correspondng symbol n the bvarate case. However, we have the followng: Theorem 5. Let S be a bvarate subdvson scheme wth a compactly supported mask correspondng to ts Laurent polynomal aðz ; z Þ¼ P ;Z a z z. Then we have () for =,, a(z,z ) has + z as a factor f and only f aðz ; z Þ ¼ 0; z¼ () a(z,z ) has + z z as a factor f and only f aðz ; t=z Þ t¼ ¼ 0 equvalently or aðz ; t=z Þ t¼ ¼ 0: ð3þ ðþ Proof. The proof s straghtforward and we show only that a(,z ) = 0 f and only f a(z,z ) has z + as a factor. We can expand a(z,z ) wth respect to z as aðz ; z Þ¼ a ðz Þz Z for some polynomals a n one varable. Then t s easy to see that a(,z ) = 0 f and only f for every Z, a ( ) = 0, whch mples that a(z,z )hasz + as a factor. The remans are shown n the same argument, whch completes the proof. h As we can see n the matrx form of the butterfly scheme below, Theorem 5 means n the geometrcal pont of vew that when we plot the masks a at the pont (,) nz -plane, the condton a(,z ) = 0 f and only f the sums of even masks and of odd masks along each horzontal lne are the same, that s to say, for every k Z, Z ð Þ a ;k ¼ 0: And we can see that a(z,t/z ) t= = 0 f and only f ð Þ a ;þk ¼ 0; k Z: Z For ntegers k and, expandng a(z,z )as aðz ; z Þ¼ Z b ðz k z Þz wth polynomals b n one varable, we have, n general, that a(z,z ) has þ z k z as a factor f and only f the mask {a } satsfes ð Þ a k; þ ¼ 0 for every Z: Z Comparng Theorem n the unvarate case, Dyn [4] found the followng crterons for the verfcaton of the convergence and smoothness of a bvarate subdvson scheme.

10 68 K.P. Ko et al. / Appled Mathematcs and Computaton 87 (007) Theorem 6 [4]. Let S be a bvarate subdvson scheme wth ts symbol a(z,z ) havng ( + z )( + z )( + z z ) as a factor. Then, S s convergent f and only f the schemes correspondng to the symbols a ;0 ðz ; z Þ¼ aðz ; z Þ ; a 0; ðz ; z Þ¼ aðz ; z Þ ; a ; ðz ; z Þ¼ aðz ; z Þ ð4þ þ z þ z þ z z are contractve. If any two of these schemes are contractve then the thrd s also contractve. Theorem 7 [4]. Let S be a bvarate subdvson scheme wth ts symbol a(z,z ) havng ( + z )( + z )( + z z ) as a factor. Then S generates C lmt functons f the schemes wth the symbols a,0 (z,z ), a 0, (z,z ) and a, (z,z )n(4) are convergent. If any of two of these schemes are convergent then the thrd s also convergent. Moreover, o S f 0 ¼ S ;0 D ;0 f 0 ; oz o S f 0 ¼ S 0; D 0; f 0 ; oz o þ o oz oz S f 0 ¼ S ; D ; f 0 : To check f a scheme generates smooth lmt functons wth the ad of Theorems 6 and 7, we have to assume that the symbol of a subdvson scheme s factorzable: aðz ; z Þ¼ðþz Þð þ z Þð þ z z Þbðz ; z Þ: ð5þ In the followng three sectons, we study symmetrc subdvson schemes wth factorzable symbols a(z,z )as n (5). At frst, we recall the most popular subdvson scheme n ths case, the butterfly scheme, and then we show that the butterfly scheme and the modfed butterfly scheme are rebult only usng the factorzaton property and ther stencl structures. Note that to verfy the smoothness of a symmetrc subdvson scheme, based on Theorems 6 and 7, we have only to check the contractvty of one of the two schemes a,0 (z,z ) and a 0, (z,z )n(4), regardng the symmetry property. 5.. The Dyn butterfly subdvson scheme Dyn et al. [6] ntroduced the butterfly scheme. The butterfly scheme s an extenson of the 4-pont nterpolatory subdvson scheme to the bvarate case wth topology of regular trangulaton, whch s an nterpolatory trangular subdvson scheme wth stencl of small support (see Fg. ). The mask of the butterfly scheme s symmetrc ða ; ¼ a ; ; ; ZÞ and gven as a 0;0 ¼ ; a ;0 ¼ a ;0 ¼ a ; ¼ a ; ¼ =; a ; ¼ a ; ¼ a ; ¼ w; a ; ¼ a 3; ¼ a ; ¼ a 3; ¼ a ; 3 ¼ a ;3 ¼ w Fg.. Stencl of the Dyn butterfly scheme: ; ¼, (3,4 = w), (5,6,7,8 = w).

11 K.P. Ko et al. / Appled Mathematcs and Computaton 87 (007) and zero otherwse. There are three knds of refnement rules as n (9): ; þ þ; þ w ; þ þ;þ þ; ¼ ;þ ¼ ; þ ;þ ; þ þ;þ þ w ; þ þ;þ þ w þ; þ ;þ w ; þ þ; þ ;þ þ þ;þ w ; þ ;þ þ þ; þ þ;þ w ; þ ; þ þ;þ þ þ;þ þ;þ ¼ In Fg., we plot the ndex of non-zero masks accordng to the refnement rules as follows. The mask maps n Fg. suggest why the scheme s called the butterfly subdvson scheme. We express the mask of the butterfly scheme n the matrx form: ) w w ) w w w A ¼ða Þ¼ ) w w w w : 0 ) ) w w w w ) w w w 5 3 ) w w From the matrx, we see that the mask satsfes the condtons that for any k Z, ð Þ a ;k ¼ ð Þ a k; ¼ ð Þ a ;þk ¼ 0 Z Z Z and Theorem 5 mples that the symbol of the butterfly scheme s factorzable. The symbol s wrtten as aðz ; z Þ¼ 3 3 ¼ 3 ¼ 3 a ; z z ¼ ð þ z Þð þ z Þð þ z z Þð wcðz ; z ÞÞðz z Þ ; where cðz ; z Þ¼z z þ z z 4z z 4z 4z þ z z þ z z þ 4z 4z 4z z þ z z þ z z : Gregory [7] computed an explct shape parameter value w 0 > /6 such that for 0 < w < w 0, the butterfly scheme generates C lmt functons on regular trangulatons. It s well-known that the scheme reproduces cubc polynomals for w = /6, otherwse lnear polynomals for w 5 / Symmetrc 8-pont butterfly subdvson scheme (a) Odd-Even masks (b) Even-Odd masks (c) Odd-Odd masks Fg.. The mask maps of the butterfly scheme. In ths secton, we consder an 8-pont symmetrc nterpolatory bvarate subdvson scheme defned on a regular trangulaton mesh. ; ; : ð6þ

12 60 K.P. Ko et al. / Appled Mathematcs and Computaton 87 (007) If we set the stencl of the symmetrc 8-pont scheme such as Fg. 3. The mask of such a scheme s gven n the matrx form as follows: ) c c ) c b c A ¼ða Þ¼ ) c b a a b c : 0 ) a a ) c b a a b c ) c b c 5 3 ) c c The bvarate Laurent polynomal of ths scheme s assumed to be factorzable: aðz ; z Þ¼ 3 3 ¼ 3 ¼ 3 a ; z z ¼ð þ z Þð þ z Þð þ z z Þbðz ; z Þ: By Theorem 5, the factorzaton mples that for each k Z, we have ð Þ a ;k ¼ ð Þ a k; ¼ ð Þ a ;þk ¼ 0: Z Z Z In ths case, we get c + b = a + = 0. If we set c = w, we obtan the same mask of the butterfly scheme: a ¼ ; b ¼ w; c ¼ w: Fg. 3. Stencl of 8-pont butterfly scheme: (, = a), (3,4 = b), (5,6,7,8 = c) Symmetrc 0-pont butterfly subdvson scheme As shown n the prevous secton, the butterfly scheme generates C surfaces n the topology of regular settng. The smoothness of surface n geometrc modelng s requred to be up to C. To obtan a subdvson Fg. 4. Stencl of the symmetrc 0-pont butterfly scheme: (, = a), (3,4 = b), (5,6,7,8 = c), (9,0 = w).

13 scheme retanng the smplcty of the butterfly scheme and creatng C lmt surfaces, we need to enlarge the support of the butterfly scheme, as mentoned n the ntroducton. We try t by takng two more ponts nto account to calculate a new control ponts as shown n Fg. 4. Comparng the stencl of the butterfly scheme n Fg. 3, we consder two nearby ponts (ponts 9 and 0 n Fg. 4) wth dfferent mask value (weght) from those of the other eght ponts. The mask of 0-pont butterfly scheme can s wrtten n the matrx form: ) w c c w ) c b c A :¼ ) c b a a b c : 0 ) w a a w ) c b a a b c ) c b c 5 3 ) w c c w We assume a factorzaton of the Laurent polynomal of ths scheme: aðz ; z Þ¼ðþz Þð þ z Þð þ z z Þcðz ; z Þ: From the factorzaton, we get c þ b ¼ 0; a w þ ¼ 0: Therefore, we fnd the mask of 0-pont butterfly scheme: a ¼ w; b ¼ c; 5; 6; 7; 8 ¼ c; 9; 0 ¼ w: Ths mask s exact wth a modfed butterfly scheme when c = /6 + w. Zorn et al. [] examned that the butterfly scheme exhbts undesrable artfacts n the case of an rregular topology, and derved an mproved scheme (a modfed butterfly scheme), whch retans the smplcty of the butterfly scheme. The modfed butterfly scheme s nterpolatng and results n smoother surfaces. References K.P. Ko et al. / Appled Mathematcs and Computaton 87 (007) [] A. Cavaretta, W. Dahmen, C.A. Mcchell, Statonary subdvson, Mem. Amer. Math. Soc. 93 (99) 86. [] G. Deslaurers, S. Dubuc, Symmetrc teratve nterpolaton processes, Constr. Approx. 5 (989) [3] N. Dyn, Interpolatory subdvson schemes, n: A. Iske, E. Quak, M.S. Floater (Eds.), Tutorals on Multresoluton n Geometrc Modellng, Sprnger, 00, pp (Chapter ). [4] N. Dyn, Analyss of convergence and smoothness by the formalsm of Laurent polynomals, n: A. Iske, E. Quak, M.S. Floater (Eds.), Tutorals on Multresoluton n Geometrc Modellng, Sprnger, 00, pp (Chapter 3). [5] N. Dyn, Subdvson schemes n computer-aded geometrc desgn, n: W.A. Lght (Ed.), Advances n Numercal Analyss II: Wavelets, Subdvson Algorthms and Radal Bass Functons, Oxford Unversty Press, 99, pp [6] N. Dyn, J.A. Gregory, D. Levn, A butterfly subdvson scheme for surface nterpolaton wth tenson control, ACM Trans. Graphcs 9 () (990) [7] J. Gregory, An Introducton to bvarate unform subdvson, n: D.F. Grffths, G.A. Watson (Eds.), Numercal Analyss, Ptman Research Notes n Mathematcs, 99, pp [8] B. Han, Analyss and constructon of optmal multdmensonal borthogonal wavelets wth compact support, SIAM J. Math. Anal. 3 (000) [9] B. Han, R.Q. Ja, Multvarate refnement equatons and convergence of subdvson schemes, SIAM J. Math. Anal. 9 (5) (998) [0] M.F. Hassan, I.P. Ivrssmtzs, N.A. Doggson, M.A. Sabn, An nterpolatng 4-pont C ternary statonary subdvson scheme, Comput. Aded Geomet. Des. 9 (00) 8. [] K.P. Ko, B.G. Lee, Y. Tang, G.J. Yoon, General formula for the mask of (n + 4)-pont symmetrc subdvson scheme, JCAM, submtted for publcaton. [] D. Zorn, P. Schröder, W. Sweldens, Interpolatng subdvson for meshes wth arbtrary topology, n: Computer Graphcs Proceedng (SIGGRAPH 96), 996, pp