1 Linear and Nonlinear Subdivision Schemes in Geometric Modeling

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1 1 Lnear and Nonlnear Subdvson Schemes n Geometrc Modelng Nra dyn School of Mathematcal Scences Tel Avv Unversty Tel Avv, Israel e-mal: nradyn@post.tau.ac.l Abstract Subdvson schemes are effcent computatonal methods for the desgn, representaton and approxmaton of 2D and 3D curves, and of surfaces of arbtrary topology n 3D. Subdvson schemes generate curves/surfaces from dscrete data by repeated refnements. Whle these methods are smple to mplement, ther analyss s rather complcated. The frst part of the paper presents the classcal case of lnear, statonary subdvson schemes refnng control ponts. It revews unvarate schemes generatng curves and ther analyss. Several well known schemes are dscussed. The second part of the paper presents three types of nonlnear subdvson schemes, whch depend on the geometry of the data, and whch are extensons of unvarate lnear schemes. The frst two types are schemes refnng control ponts and generatng curves. The last s a scheme refnng curves n a geometry-dependent way, and generatng surfaces. 1.1 Introducton Subdvson schemes are effcent computatonal tools for the generaton of functons/curves/surfaces from dscrete data by repeated refnements. They are used n geometrc modelng for the desgn, representaton and approxmaton of curves and of surfaces of arbtrary topology. A lnear statonary scheme uses the same lnear refnement rules at each locaton and at each refnement level. The refnement rules depend on a fnte number of mask coeffcents. Therefore, such schemes are easy to mplement, but ther analyss s rather complcated. The frst subdvson schemes where devsed by G. de Rahm (1956) 1

2 2 Nra Dyn for the generaton of functons wth a frst dervatve everywhere and a second dervatve nowhere. Infact, n most cases, the lmts generated by subdvson schemes are convolutons of a smooth functon wth a fractal one. Ths paper conssts of two parts. The frst s a revew of classcal subdvson schemes, namely lnear, statonary schemes refnng control ponts. The second brngs several constructons of nonlnear schemes, takng nto account the geometry of the refned obects. The nonlnear schemes are all extensons of good lnear unvarate schemes. In the frst part we revew only lnear unvarate subdvson schemes (schemes generatng curves from ntal ponts). Although the man applcaton of subdvson schemes s n generatng surfaces, we lmt the presentaton here to these schemes. The man reasons for ths choce are two. The theory of unvarate schemes s much more complete and easer to present, yet t conssts of most of the aspects of the theory of schemes generatng surfaces. The understandng of ths theory s a frst essental step towards the understandng of the theory of schemes generatng surfaces. Also, the schemes that are extended n the second part to nonlnear schemes, are unvarate. Secton 1.2 s devoted to the presentaton of statonary, lnear, unvarate subdvson schemes. Important examples such as the B-splne schemes and the nterpolatory 4-pont scheme are dscussed n detals. Among these mportant examples are the schemes that are extended n the second part to nonlnear schemes. A sketch of tools for analyss of convergence and smoothness of lnear, unvarate schemes s gven n The relaton between subdvson schemes and the constructon of wavelets s brefly dscussed n The second part of the paper conssts of three sectons. In Secton 1.3 lnear schemes are extended to refne manfold-valued data. Ths s done n two steps. Frst, the refnement rules of any convergent lnear scheme are presented (n several possble ways) n terms of repeated bnary averages. Ths s demonstrated by several examples n Then n the manfold-valued subdvson schemes are constructed, ether by replacng every lnear bnary average n the lnear refnement rules by a geodesc average, yldng a geodesc analoguous scheme, or by replacng every lnear bnary average n the lnear refnement rules by ts proecton to the manfold, yeldng a proecton analogue scheme. The analyss of the so constructed manfold-valued subdvson schemes, s done n 1.3.3, va ther proxmty to the lnear schemes from whch they are derved.

3 In Secton 1.4 two data-dependent extensons of the lnear nterpolatory 4-pont scheme are dscussed. In both extensons the refnement s adapted to the local geometry of the four ponts used. These two geometrc (data-dependent) 4-pont schemes are effectve n case of an ntal control polygon wth edges of sgnfcantly dfferent length. For such ntal control polygons the lmts generated by the lnear 4-pont scheme have unwanted features (artfacts), whle the geometrc 4-pont schemes tend to generate artfact free lmts. The last secton deals wth repeated refnements of curves for the generaton of a surface. Here the scheme that s extended s the quadratc B-splne scheme (called Chakn algorthm). It s used to refne curves by frst constructng a geometrc correspondence between pars of curves, and then applyng the refnment to correspondng ponts. Snce ths s a work n progress only partal results are presented Lnear subdvson schemes for the generaton of curves In ths secton we dscuss statonary, lnear schemes, generatng curves by repeated refnements of ponts. Such a subdvson scheme s defned by a fnte set of real coeffcents called mask a = {a R, σ(a) Z}, where σ(a) denotes the fnte support of the mask. The scheme wth the mask a s denoted by S a. The refnement rules of S a have the form (S a P) α = β Za α 2β P β, α Z, (1.1) where P denotes the polygonal lne through the ponts {P } Z R d, wth d 1. Note that there are two dfferent refnement rules n (1.1), one correspondng to odd α and one to even α, nvolvng the odd respectvely even coeffcents of the mask. Remark 1.1 In ths secton we consder schemes operatng on ponts defned on Z, although, n geometrc applcatons the schemes operate on fnte sets of ponts. Due to the fnte support of the mask, our consderatons apply drectly to closed curves and also to open ones, except n a fnte zone near the boundary.

4 4 Nra Dyn The ntal ponts to be refned are called control ponts, and the correspondng polygonal lne through them s called control polygon. These terms are used also for the ponts and the correspondng polygonal lne at each refnement level. The subdvson scheme s a sequence of repeated refnements of control polygons, {P k+1 = S a P k, k 0}, (1.2) wth P 0 the ntal control polygon. The refnements n (1.2) are statonary, snce at each refnement level the same scheme S a s used. Materal on nonstatonary subdvson schemes can be found n the revew paper Dyn & Levn (2002). A subdvson scheme s termed unformly convergent f the sequence of control polygons {P k } converges unformly on compact sets of R d. Ths s the noton of convergence relevant to geometrc applcatons. Each control polygons n (1.2) has a parametrc representaton as the vector n R d of the pecewse lnear nterpolant to the data {(2 k, P k), Z}, and the unform convergence s analyzed n the settng of contnuous functons. (see e.g. Dyn (1992)). For P 0 R d wth d = 1, the scheme converges to a unvarate functon, whle n case d 2 the scheme converges to a curve n R d. The convergence of a scheme S a mples the exstence of a basc-lmtfuncton φ a, whch s the lmt obtaned from the ntal data, δ 0 = 0 everywhere on Z except δ0 0 = 1. It follows from the lnearty and unformty of (1.1) that the lmt SA P0, obtaned from any ntal control polygon P 0 passng through the ntal control ponts {Pα 0 R d, α Z}, can be wrtten n terms of nteger translates of φ a, as (S a P 0 )(t) = α Z P 0 αφ a (t α), t R. (1.3) Equaton (1.3) s a parametrc representaton of a curve n R d for d 2. Also, by the lnearty, unformty and statonarty (see (1.2)) of the subdvson scheme, and snce (S a δ 0 ) α = a α, α Z, φ a satsfes the refnement equaton (two-scale relaton) φ a (t) = α Za α φ a (2t α). (1.4) It s easy to obtan from (1.1) or from (1.4) that the support of φ a equals the convex hull of σ(a) (see e.g. Cavaretta, Dahmen & Mcchell (1991)).

5 Remark 1.2 The dscusson above leads to the concluson that a convergent subdvson scheme gves rse to a unque contnuous basc-lmtfuncton, satsfyng a refnement equaton wth the mask of the scheme. The converse s not generally true. Yet a contnuous functon satsfng a refnement equaton s a basc-lmt-functon of a convergent subdvson scheme, f ts nteger shfts are lnearly ndependent (see Cavaretta, Dahmen & Mcchell (1991)) The man types of schemes The frst subdvson schemes n geometrc modellng were proposed for easy and quck renderng of B-splne curves. A B-splne curve has the form C(t) = P B m (t ) (1.5) wth {P } R d the control ponts, and B m a B-splne of degree m wth nteger knots, namely B m [,+1] s a polynomal of degree m for each Z, B m C m 1 (R), and B m has a compact support [0, m + 1]. Equaton (1.5) s a parametrc representaton of a B-splne curve. The B-splne curves (1.5) are a powerful desgn tool, snce ther shape s smlar to the shape of the control polygon P correspondng to the control ponts n (1.5) (see e.g. Prautzsch, Boehm & Paluszny (2002)). Such curves can be well approxmated by the control polygons generated by the repeated refnements (1.2), usng the mask a [m] wth coeffcents ( ) a [m] m + 1 = 2 m, = 0,...,m + 1. (1.6) The repeated refnements of a B-splne scheme of degree m are thus P l+1 = a [m] 2 P l, Z, l = 0, 1, 2,.... (1.7) In (1.7) we use the conventon a [m] = 0, / {0, 1,..., m + 1}. By the convergence analyss, presented n 1.2.2, t s easy to check that the subdvson scheme wth the mask (1.6) s convergent. It converges to B-splne curves of degree m, snce B m s the basc-lmtfuncton of S a [m]. Ths can be easly concluded n vew of Remark 1.2 from the defnton of B-splnes, by observng that B m satsfes a refnement equaton ( 1.4)

6 6 Nra Dyn wth the mask a [m] (see e.g. Dyn (1992)), and that the nteger translates of B m are lnearly ndependent. Thus the control polygon P k = S k ap approxmates C(t) for k large enough, and C(t) can be easly rendered by renderng ts approxmaton P k. In practce a large enough k s about 4 as can be seen n Fgure 1.1. The frst scheme of ths type was devsed n Chakn (1974) for a fast geometrc renderng of quadratc B-splne curves. It has the refnement rules P k+1 2 = 3 4 P k P k, P k = 1 4 P k P k. (1.8) Fgure 1.1 llustrates three refnement steps wth ths scheme, appled to a closed ntal control polygon. Chakn scheme s extended to nonlnear schemes n 1.3 and n 1.5. orgnal teraton #1 teraton #2 teraton #3 Fg Refnements of a polygon wth Chakn scheme The schemes for general B-splne curves were ntroduced and nvestgated n Cohen, Lyche & Resenfeld (1980). All other subdvson schemes can be regarded as a generalzaton of the B-splne schemes. The B-splne schemes generate curves wth a smlar shape to the shape of the ntal control polygons, but do not pass through the ntal control ponts. Schemes wth lmt curves that nterpolate the ntal control ponts, were ntroduced n the late 1980 s. These schemes are called nterpolatory, and have the even refnement rule P k+1 2 = P k. Thus the control ponts at refnement level k are contaned n those at refnement level k + 1. The odd refnement rule, called n case of

7 nterpolatory schems nserton rule, s desgned by a local approxmaton based on nearby control ponts. The frst schemes of ths type were ntroduced by Deslaurers & Dubuc (1988) and by Dyn, Gregory & levn (1987). In the frst paper the nserton rule for the pont P k s obtaned by frst nterpolatng the data {(( + ), P+ k ), = N + 1,...,N} by a vector polynomal of degree 2N 1, and then samplng t at the pont Ths s done for a fx postve N. Rgardng N as a parameter, ths constructon yelds a one-parameter famly of convergent nterpolatory subdvson schemes, wth masks of ncreasng support, and wth basc-lmt-functons of ncreasng smoothness (see also the book Daubeches (1992)). We denote the scheme n ths famly correspondng to N by DD N. In the second paper above, a one-parameter famly of 4-pont nterpolatory schemes s ntroduced, wth the nserton rule P k = w(p k 1 + P k +2) + ( w)(p k + P k +1). (1.9) Here w s a shape parameter. For w = 0 the lmt s the ntal control polygon, whle as w ncreases to w = 1/8 the lmt s a C 1 curve whch becomes looser relatve to the ntal control polygon, as demonstrated n Fgure 1.3. Thus w acts as a tenson parameter. For w = 1 16 ths scheme concdes wth DD 2. The local approxmaton whch s used n the constructon of the nserton rule (1.9) s a convex combnaton of the cubc nterpolant used n DD 2 and the lnear nterpolant used n DD 1. In the next subsecton the dependence of the convergence and smoothness of the 4-pont scheme on the parameter w s dscussed. More about general nterpolatory subdvson schemes, ncludng multvarate schemes, can be found n Dyn & Levn (1992). Whle n case of the B-splne schemes the lmt was known and convergence was guaranteed, n case of the nterpolastory schemes, analyss tools had to be developed. The analyss n Deslaurers & Dubuc (1988) and Daubeches (1992) and n references theren s manly n the Fourer doman, whle n Dyn, Gregory & levn (1987) t s done n the geometrc doman, based on the symbol of the scheme. Hnts on ths method of analyss, whch was further developed n Dyn, Gregory & Levn (1991), are gven n the next subsecton. 7

8 8 Nra Dyn Analyss of convergence and smoothness Gven the coeffcents of the mask of a scheme, one would lke to be able to determne f the scheme s convergent, and what s the smoothness of the resultng basc lmt functon (whch s the generc smoothness of the lmts generated by the scheme n vew of (1.3)). Such analyss tools are also essental for the desgn of new schemes. We sketch here the method of convergence and smoothness analyss n Dyn, Gregory & Levn (1991) (see also Dyn (1992)). An mportant tool n the analyss s the symbol of a scheme S a wth the mask a = {a α : α σ(a)}, a(z) = a α z α. (1.10) α σ(a) In the followng we use also the notaton S a(z) for S a. A frst step towards the convergence analyss s the dervaton of the necessary condton for unform convergence, a α 2β = 1, α = 0 or 1 (mod 2), (1.11) β Z The condton n (1.11) s derved easly from the refnement step (S k+1 a P) α = β Za α 2β (S k ap) β, α Z, for k large enough so that for all l k, S l a P S a P s small enough. The necessary condton (1.11) mples that we have to consder symbols satsfyng Condton (1.12) s equvalent to a(1) = 2, a( 1) = 0. (1.12) a(z) = (1 + z)q(z) wth q(1) = 1. (1.13) The scheme wth symbol q(z), S q, satsfes S q = S a, where s the dfference operator P = { ( P) = P P 1, Z }. (1.14) A necessary and suffcent condton for the convergence of S a s the contractvty of the scheme S q, namely S a s convergent f and only f S q P 0 for any P. The contractvty of S q s equvalent to the exstence of a postve nteger L, such that S L q < 1. Ths condton can be checked for a gven L by algebrac operatons on the symbol q(z).

9 For practcal geometrcal reasons, only small values of L have to be consdered, snce a small value of L guarantees vsual convergence of {P k } to Sa P 0, already for small k, as the dstances between consecutve control ponts contract to zero fast. A good scheme corresponds to L = 1 as the B-splne schemes, or to L = 2 as the 4-pont scheme. The smoothness analyss reles on the result that f the symbol of a scheme has a factorzaton ( ) ν 1 + z a(z) = b(z), (1.15) 2 such that the scheme S b s convergent, then S a s convergent and ts lmt functons are related to those of S b by D ν (S a P) = S b ν P, (1.16) wth D the dfferentaton operator. Thus, each factor (1 + z)/2 multplyng a symbol of a convergent scheme adds one order of smoothness. Ths factor s termed smoothng factor. The relaton between (1.15) and (1.16) s a partcular nstance of the algebra of symbols (see e.g. Dyn & Levn (1995)). If a(z), b(z) are two symbols of convergng schemes, then S c wth the symbol c(z) = 1 2 a(z)b(z) s convergent, and φ c = φ a φ b. (1.17) 9 Example (B-splne schemes). In vew of (1.6), the symbol of the scheme generatng B-splne curves of degree m s a(z) = (1 + z) m+1 /2 m. (1.18) The known smoothness of the lmt functons generated by the m-th degree B-splne scheme, can be concluded easly, usng the tools of analyss presented n ths subsecton. The factor b(z) = (1+z) 2 2 corresponds to the scheme S b generatng the ntal control polygon as the lmt curve, whch s contnuous, whle the factors ( ) 1+z m 1 2 add smoothness, so that S P 0 C m 1. a [m] Example (the 4-pont scheme). The analyss sketched above, s the tool by whch the followng results were obtaned for the nterpolatory 4-pont scheme wth the nserton rule (1.9). The symbol of the scheme can be wrtten as a w (z) = 1 2z (z + 1)2[ 1 2wz 2 (1 z) 2 (z 2 + 1) ]. (1.19)

10 10 Nra Dyn The range of w for whch S aw(z) s convergent s the range for whch S qw(z) wth symbol q w (z) = a w (z)/(1 + z) s contractve. The condton S qw(z) < 1 holds n the range 3/8 < w < ( )/8, whle the condton S 2 q w(z) < 1 holds n the range 1/4 < w < ( )/8. Thus S aw(z) s convergent n the range 3/8 < w < ( )/8. In fact t was shown by M.D. Powell that the convergence range s w 1 2. To fnd a range of w where S aw(z) generates C 1 lmts, the contractvty of S cw(z) wth c w (z) = 2a w (z)/(1 + z) 2 has to be nvestgated. It s easy to check that S cw(z) 1, but that S 2 c w(z) < 1 for 0 < w < ( 5 1)/8. Only a year ago the maxmal postve w for whch the lmt s C 1 was obtaned n Hechler, Mößner & Ref (2008). The lmt of S aw(z) s not C 2 even for w = 1/16, although for w = 1/16 the symbol s dvsble by (1+z) 3. It s shown n Daubeches & Lagaras (1992), by other methods, that the basc lmt functon for w = 1/16, restrcted to ts support, has a second dervatve only at the non-dyadc ponts there. The condtons for smoothness gven above are only suffcent. Yet, there s a large class of convergent schemes for whch the factorzaton n (1.15) s necessary for generatng C ν lmt functons. Ths class contans the B-splne schemes and the nterpolatory schemes. See e.g. Dyn & Levn (2002) Subdvson schemes and the constructon of wavelets Any convergent subdvson scheme S a defnes a sequence of nested spaces n terms of ts basc-lmt-functon φ a. For every k Z defne the space V k = span{φ a (2 k ( )), 2 k Z} = span{φ a (2 k ), Z}. (1.20) Then n vew of the refnement equaton (1.4) satsfed by φ a, these spaces are nested, namely... V 1 V 0 V 1 V 2... (1.21) Such a b-nfnte sequence of spaces s the framework n whch wavelets are constructed. It s called a multresoluton analyss when the nteger translates of φ a consttute a Resz bass of V 0 (see Daubeches (1992)). In the wavelets lterature one starts from a soluton of (1.4), termed scalng

11 functon, whch s not necessarly the lmt of a convergng subdvson scheme, as ndcated by Remark 1.2. The chose of the mask coeffcents for the constructon of wavelets depends on the propertes requred from the wavelets. For example the orthonormal wavelets of Daubeches are generated by masks (flters) that are related to the masks of the DD N schemes. The symbols of these masks, denoted by ã N (z), satsfy ã N (z) 2 = a N (z), wth a N (z) the symbol of the scheme DD N. Ths relaton between the symbols can be expressed n terms of the correspondng scalng functons as φãn φãn ( ) = φ an, showng that the nteger translates of φãn consttute an orthonormal system due to the nterpolatory nature of φ an, whch vanshes at all ntegers except at zero where t s 1. The next sectons brng several constructons of non-lnear schemes, based on lnear schemes. In more nformaton on lnear schemes, needed for the constructon of schemes on manfolds, s presented Curve subdvson schemes on manfolds To desgn subdvson schemes for curves on a manfold, we requre that the control ponts generated at each refnement level are on the manfold, and that the lmt of the sequence of correspondng control polygons s on the manfold. Such schemes are nonlnear. The frst approach to ths problem s presented n Rahman, Dror, Stodden, Donoho & Schröder (2005). It s based on adaptng a lnear unvarate subdvson scheme S a. Gven control ponts {P } on the manfold, let P = {P } denote the correspondng control polygon, and let T denote the adaptaton of S a to the manfold. Then the pont (T P) s defned by frst executng the lnear refnement step of S a on the proectons of the ponts {P, a 2 0} to a tangent plane at a chosen pont P, and then proectng the obtaned pont to the manfold. Ths can be wrtten as (T P) = ψ 1 P ( Z ) a 2 ψ P (P ), (1.22) where P s some chosen center of the ponts {P, a 2 0}, and ψ P s the proecton from the manfold to the tangent plan at P. Recently t was shown n Xe & Yu (2008) and Xe & Yu (2008a), that wth a proper chose of the center pont many propertes of the lnear

12 12 Nra Dyn scheme, such as convergence, smoothness and approxmaton order, are shared by the nonlnear scheme derved from t. Here we dscuss two other constructons of subdvson schemes on manfolds from convergng lnear schemes. These constructons are based on the observaton that the refnement rules of any convergent lnear scheme can be calculated by repeated bnary averages (see Wallner & Dyn (2005)) Lnear schemes n terms of repeated bnary averages A lnear scheme for curves, S, s defned by two refnement rules of the form, (SP) = a 2 P, = 0 or 1(mod 2), (1.23) where P = {P }. As dscussed n 1.2.2, any convergent lnear scheme s affne nvarant, namely a 2 = 1. It s shown n Wallner & Dyn (2005), that for a convergent lnear scheme, each of the refnement rules n (1.23) s expressble, n a non-unque way, by repeated bnary averages. A reasonable choce s a symmetrc representaton relatve to the topologcal relatons n the control polygon. For example, wth the notaton Av α (P, Q) = (1 α)p + αq, α R, P, Q R d, the nserton rule of the nterpolatory 4-pont scheme (1.9) can be rewrtten as ( ) P2+1 k = Av Av1 2 ( 2w) (P, P 1 ), Av ( 2w) (P +1, P +2 ), or as ) P2+1 k = Av ( 2w) (Av1 (P, P 2 +1 ), Av1 (P 1, P 2 +2 ). Refnement rules represented n ths way are termed hereafter refnement rules n terms of repeated bnary averages. Among the lnear schemes there s a class of factorzable schemes for whch the symbol a(z) = a z, can be wrtten as a product of lnear real factors. For such a scheme, the control polygon obtaned by one refnement step of the form (1.1), can be computed by several smple global steps, unquely determned by the factors of the symbol. To be more specfc, let us consder a symbol of the form a(z) = (1 + z) 1 + x 1z 1 + x x mz 1 + x m. (1.24) Note that ths symbol corresponds to an affne nvarant scheme snce

13 a(1) = 2, and a( 1) = 0. In fact, the form of the symbol n (1.24) s general for convergng factorzable schemes. Let P k denote the control polygon at refnement level k, correspondng to the control ponts {P k }. Each smple step n the executon of the refnement S a P k corresponds to one factor n (1.24). The frst step n calculatng the control ponts at level k + 1 corresponds to the factor 1 + z, and conssts of elementary refnement: P k+1, = P k+1,0 2+1 = P k. (1.25) Ths step s followed by m averagng steps correspondng to the factors 1+x z 1+x, = 1,...,m. The averagng step dctated by the factor 1+xz 1+x s, P k+1, = 1 (P k+1, 1 + x P k+1, 1 1 ), Z. (1.26) 1 + x The control ponts at level k + 1 are P k+1 = P k+1,m, Z. The executon of the refnement step by several smple global steps s equvalent to the observaton that S a(z) P k = R 1+x 1 z R 1+xmz S (1+z) P k, (1.27) 1+x 1 1+xm wth (R b+cz P) = bp + cp 1. The equalty (1.27) follows from the representaton of (1.1) as the formal equalty (S a P) z = a(z) P z 2, (1.28) wth P = {P }. The formal equalty n (1.28) s n the sense of equalty between the coeffcents of the same power of z n both sdes of (1.28). We term the executon of the refnement step by several smple global steps, based on the factorzaton of the symbol to lnear factors, global refnement procedure by repeated averagng. An mportant famly of factorzable schemes s that of the B-splne schemes, wth symbols gven by (1.18). Note that the factors n (1.18) correspondng to repeated averagng are all of the form 1+z 2. Thus the global refnement procedure by repeated averagng s equvalent to the algorthm of Lane & Resenfeld (1980). Also, t follows from the fact that all factors n (1.18) except for the factor 1 + z are smoothng factors, that any B-splne scheme s optmal, n the sense that t has a mask of mnmal support among all schemes wth the same smoothness. The nterpolatory 4-pont scheme gven by the nserton rule (1.9) wth

14 14 Nra Dyn Fg Geodesc B-Splne subdvson of degree three. From left to rght: Tp, T 2 p,t 3 p, T p. w = 1 16 (whch s the DD 2 scheme) s also factorzable. Its symbol has the form (see e.g. Dyn (2005)) a(z) = (1 + z) 4 (3 + 3z)(3 3z)/48. (1.29) Constructon of subdvson schemes on manfolds The two constructons of nonlnear schemes on manfolds n Wallner & Dyn (2005), start from a convergent lnear scheme, S, gven ether by local refnement rules n terms of repeated bnary averages, or by a global refnement procedure n terms of repeated bnary averages. The frst constructon of a subdvson T on a manfold M, analogous to S, replaces every bnary average n the representaton of S, by a correspondng geodesc average on M. Thus Av α (P, Q) s replaced by gav α (P, Q), where gav α (P, Q) = c(ατ), wth c(t) the geodesc curve on M from P to Q, satsfyng c(0) = P and c(τ) = Q. The resultng subdvson scheme s termed geodesc subdvson scheme. The second constructon uses a smooth proecton mappng onto M, and replaces every bnary average by ts proecton onto M. The resultng nonlnear scheme s termed a proecton subdvson scheme. In case of a surface n R 3, a possble choce of the proecton mappng s the orthogonal proecton onto the surface. Note that for a factorzable scheme the analoguous manfold schemes obtaned from ts representaton n terms of the global refnement procedure by repeated averagng, depend on the order of the lnear factors correspondng to bnary averages n (1.24). Yet for the B-splne schemes there s one geodesc analoguous scheme, and one proecton analogous scheme obtaned from ths representaton, snce all the factors n (1.18), except the factor (1 + z), are dentcal. Example. In ths example the lnear scheme s the Chakn algorthm

15 15 of (1.8) P k+1 2 wth the symbol = Av1 4 (P k 1, P k ), P k = Av3 4 (P k 1, P k ), (1.30) a(z) = (1 + z) 3 /4. (1.31) The dfferent adaptatons to the manfold case are: () Chakn geodesc scheme, derved from (1.30): P k+1 2 = gav1 4 (P k 1, P k ), P k = gav3 4 (P k 1, P k ). () Chakn geodesc scheme, derved from (1.31): P k+1,0 2 = P k+1,0 2+1 = P k, P k+1, () Chakn proecton scheme derved from (1.30): P k+1 2 k+1, 1 = gav1 (P 2, P k+1, 1 1 ), = 1, 2. = G(Av1 4 (P k 1, P k )), P k = G(Av3 4 (P k 1, P k )). (v) Chakn proecton scheme derved from (1.31): P k+1,0 2 = P k+1,0 2+1 = P k, P k+1, k+1, 1 = G(Av1 (P 2, P k+1, 1 1 ), = 1, 2. In the above G s a specfc smooth proecton mappng to the manfold M. Fgure 1.2 dsplays a curve on a sphere, created from a fnte number of ntal control ponts on the sphere, by a geodesc analoguous scheme to a thrd degree B-splne scheme Analyss of convergence and smoothness by proxmty The analyss of convergence and smoothness of the geodesc and the proecton schemes we present, s based on ther proxmty to the lnear scheme from whch they are derved, and on the smoothness propertes of ths lnear scheme. We lmt the dscusson to C 1 and C 2 smoothness. To formulate the proxmty condtons we ntroduce some notaton. For a control polygon P = {P }, we defne P = {P P 1 }, l P = ( l 1 P), d l (P) = max ( l P). (1.32) The dfference between two control polygons P = {P }, Q = {Q }, s P Q = {P Q }. Wth ths notaton the two proxmty relatons of nterest to us are the followng:

16 16 Nra Dyn Defnton 1.1 () Two schemes S and T are n 0-proxmty f d 0 (SP T P) Cd 1 (P) 2, for all control polygons P wth d 1 (P) small enough. () Two schemes S and T are n 1-proxmty f d 1 (SP T P) C[d 1 (P)d 2 (P) d 1 (P) 3 ], for all control polygons P wth d 1 (P) small enough. In these defntons C s a generc constant. From the 0-proxmty condton we can deduce the convergence of T from the convergence of the lnear scheme S. Furthermore, under mld condtons on S, we can also deduce that f S generates C 1 lmt curves then T generates C 1 lmt curves whenever t converges. In Wallner (2006), results on C 2 smoothness of the lmt curves generated by T are obtaned, based on 0-proxmty and 1-proxmty of T and S, and some mld condtons on S, n addton to S beng C 2. The 0-proxmty and the 1-proxomty condtons hold for the manfold schemes of subsecton 1.3.2, when S s a convergent lnear subdvson scheme and when M s a smooth manfold. Moreover, for M a compact manfold or a surface wth bounded normal curvatures, the two proxmty condtons hold unformly for all P such that d 1 (P) < δ, wth a global δ. Examples. The B-splne schemes wth symbol (1.18) for m 3, generate C 2 curves and satsfy the mld condtons necessary for deducng that the lmt curves of ther manfold analoguous schemes are also C 2. On the otherhand the lnear 4-pont scheme generates lmt curves whch are only C 1, a property that s shared by ts manfold analoguous schemes. Further analyss of manfold-valued subdvson schemes can be found n a seres of papers by Wallner and hs collaborators, see e.g. Wallner, Nava Yazdan & Grohs (2007), Grohs & Wallner (2008), and n the works Xe & Yu (2007), Xe & Yu (2008), Xe & Yu (2008a). 1.4 Geometrc 4-pont nterpolatory subdvson schemes The refnement step of a lnear scheme (S a P) = a 2P, Z, s appled separately to each component of the control ponts. Therefore these schemes are nsenstve to the geometry of the control polygons.

17 For control polygons wth edges of smlar length, ths nsenstvty s not problematc. Yet, the lmt curves generated by lnear schemes, n case the ntal control polygon has edges of sgnfcantly dfferent length, have artfacts, namely geometrc features whch do not exst n the ntal control polygon. Ths can be seen n the upper rght fgure of Fgure 1.3 and n the second column of Fgure 1.4. Data dependent schemes can cure ths problem. Here we present two geometrc versons of the 4-pont scheme, whch are data dependent. The frst s based on adaptng the tenson parameter to the geometry of the 4-ponts nvolved n the nserton rule, the second s based on a geometrc parametrzaton of the control polygon at each refnement level Adaptve tenson parameter In ths secton we present a nonlnear verson of the lnear 4-pont nterpolatory scheme, ntroduced n Marnov, Dyn & Levn (2005), whch adapts the tenson parameter to the geometry of the control ponts. It s well known that the lnear 4-pont scheme wth the refnement rules P k+1 2 = P k, P k = w(p k 1 +P k +2 )+(1 2 + w)(p k + P k +1 ), (1.33) where w s a fxed tenson parameter, has the followng attrbutes: It generates good curves when appled to control polygons wth edges of comparable length. It generates curves whch become smoother (have greater Hölder exponent of the frst dervatve), the closer the tenson parameter s to 1/16. Only for very small values of the tenson parameter, t generates a curve whch preserve the shape of an ntal control polygon wth edges of sgnfcantly dfferent length. (Recall that the control polygon tself corresponds to the generated curve wth zero tenson parameter.) We frst wrte the refnement rules n (1.33) n terms of the edges {e k = P +1 k P k } of the control polygon, and relate the nserted pont P2+1 k+1 to the edge ek. The nserton rule can be wrtten n the form, P e k = M e k + w e k (e k 1 e k +1) (1.34)

18 18 Nra Dyn Fg Curves generated by the lnear 4-pont scheme: (Upper left) the effect of dfferent tenson parameters, (upper rght) artfacts n the curve generated wth w = 1, (lower left) artfact-free but vsually non-smooth curve 16 generated wth w = Artfact-free and vsually smooth curve generated n a nonlnear way wth adaptve tenson parameters (lower rght). wth M e k the mdpont of e k and w e k the adaptve tenson parameter. Defnng d e k = w e k (e k 1 ek +1 ) as the dsplacement from M e k, we control ts sze by choosng w e k accordng to a geometrcal crteron. In Marnov, Dyn & Levn (2005) there are varous geometrcal crtera, all of them guaranteeng that the nserted control pont P e k s dfferent from the boundary ponts of the edge e k, and that the length of each of the two edges replacng e k s bounded by the length of ek. Ths s acheved f w e k s chosen so that d e k 1 2 ek. (1.35) All the crtera restrct the value of the tenson parameter w e k to the nterval (0, 1 16 ], such that a tenson close to 1/16 s assgned to regular stencls namely, stencls of four ponts wth three edges of almost equal length, whle the less regular the stencl s, the closer to zero s the tenson parameter assgned to t. A natural choce of an adaptve tenson parameter obeyng (1.35) s { 1 w e k = mn 16, c e k } e k 1 ek +1, wth a fxed c [ 1 8, 1 2 ). (1.36) 1 In (1.36) c s restrcted to the nterval [ 1 8, 1 2 ) to guarantee that w e k = 16 for stencls wth ek 1 = ek = ek +1. Indeed n ths case, ek 1 e k +1 = 2 sn θ 2 ek, wth θ, 0 θ π, the angle between the two vectors e k 1, ek +1. Thus ek / ek 1 ek +1 = (2 sn θ 2 ) 1 1 2, and

19 f c 1 8 then the mnmum n (1.36) s The choce (1.36) defnes rregular stencls (correspondng to small w e k ) as those wth e k much smaller than at least one of e k 1, ek +1, and such that when these two edges are of comparable length, the angle between them s not close to zero. The convergence of ths geometrc 4-pont scheme, and the contnuty of the lmts generated, follow from a result n Levn (1999). There t s proved that the 4-pont scheme wth varable tenson parameter s convergent, and that the lmts generated are contnuous, whenever the tenson parameters are restrcted to the nterval [0, w], wth w < 1 8. But we cannot apply the result n Levn (1999), on C 1 lmts of the 4-pont scheme wth varable tenson parameter to the geometrc 4-pont scheme defned by (1.34) and (1.36), snce the tenson parameters used durng ths subdvson process, are not bounded away from zero. Nevertheless, many smulatons ndcate that the curves generated by ths scheme are C 1 (see Marnov, Dyn & Levn (2005)) Geometrc parametrzaton of the control polygons In ths subsecton we present a geometrc 4-pont scheme, whch s ntroduced and nvestgated n Dyn, Floater & Hormann (2008). The dea for the geometrc nserton rule of the pont P2+1 k, comes from the nserton rule of the DD 2 scheme (see 1.2.1). The nserton rule of the DD 2 scheme s obtaned by samplng the vector cubc polynomal, nterpolatng the data {(( + ), P+ k ), = 1, 0, 1, 2}, at the pont From ths pont of vew, the lnear scheme corresponds to a unform parametrzaton of the control polygon at each refnement level. Ths approach fals when the ntal control polygon has edges of sgnfcantly dfferent length. Yet the use of the centrpetal parametrzaton, nstead of the unform parametrzaton, leads to a geometrc 4-pont scheme wth artfact-free lmt curves, as can be seen n Fgure 1.4. The centrpetal parametrzaton, whch s known to be effectve for nterpolaton of control ponts by a cubc splne curve (see Floater (2008)), has the form t cen (P) = {t }, wth t 0 = 0, t = t 1 + P P , (1.37) where 2 s the Eucldean norm, and P = {P }. Let P k be the control polygon at refnement level k, and let {t k } =

20 20 Nra Dyn control polygon unform centrpetal chordal Fg Comparsons between 4-pont schemes based on dfferent parametrzatons. t cen (P k ). The refnement rules for the geometrc 4-pont scheme, based on the centrpetal parametrzaton are: 1 ) = P k, P k k,( = π 2 (tk + t k +1), P k+1 2 wth π k, the vector of cubc polynomals, satsfyng the nterpolaton condtons π k, (t k +) = P k +, = 1, 0, 1, 2. Note that ths constructon can be done wth any parametrzaton. In fact n Dyn, Floater & Hormann (2008) the chordal parametrzaton (t +1 t = P +1 P 2 ) s also nvestgated, but found to be nferor to the centrpetal parametrzaton (see Fgure 1.4). In contrast to the analyss of the schemes on manfolds, the method of analyss of the geometrc 4-pont schemes, based on the chordal and centrpetal parametrzatons s rather ad-hoc. It s shown n Dyn, Floater & Hormann (2008) that both schemes are well defned, n the sense that any nserted pont s dfferent from the end ponts of the edge to whch t corresponds, and that both schemes are convergent to contnuous lmt curves. Although numercal smulatons ndcate that both schemes generate C 1 curves, as the lnear 4-pont scheme, there s no proof of such a property there. Another type of nformaton on the lmt curves, whch s relevant to the absence or presence of artfacts, s avalable n Dyn, Floater & Hormann (2008). Bounds on the Hausdorff dstance from sectons of a lmt curve to ther correspondng edges n the ntal control polygon are derved. These bounds gve a partal qualtatve understandng why the lmt curves correspondng to the centrpetal parametrzaton are artfact free.

21 Let C denote a curve generated by the scheme based on the centrpetal parametrzaton from an ntal control polygon P 0. Snce the scheme s nterpolatory, C passes through the ntal control ponts. Denote by C e 0 the secton of C startng at P 0 and endng at P+1 0. Then haus(c e 0, e 0 ) 5 7 e0 2. Thus the secton of the curve cannot be too far from a short edge. On the other hand the correspondng bound n the lnear case has the form haus(c e 0, e 0 ) 3 13 max{ e0 2, 2}, and a secton of the curve can be rather far from ts correspondng short edge, f ths edge has a long neghborng edge. In case of the chordal parametrzaton the bound s even worse haus(c e 0, e 0 ) 11 5 max{ e0 2, 2}. Comparsons of the performance of the three 4-pont schemes, dscussed n ths secton, are gven n Fgure Geometrc refnement of curves The scheme dscussed n ths secton s desgned and nvesgated n Dyn, Elber & Ita (2008). It s a nonlnear extenson of the quadratc B-splne scheme S a, correspondng to the symbol gven by (1.18) wth m = 2. S a s a lnear scheme generatng a curve from a set of ntal control ponts, yet ts extenson presented here generates surfaces, as the refned obects are not control ponts but control curves. Ths new nonlnear scheme repeatedly refnes a set of control curves, takng nto account the geometry of the curves, so as to generate a lmt surface, whch s related to the geometry of the ntal control curves. Infact, a surface can be generated from an ntal set of curves {C } n =0, usng S a n a lnear way. The ntal curves have to be parametrzed n some reasonable way to yeld the set {C (s), s [0, 1]} n =0, and then S a s appled to each control polygon of the form P s = {C (s)} n =0 correspondng to a fxed s n [0, 1]. Ths s equvalent to refnng the curves wth S a, to obtan the refned set of control curves S a P s, s [0, 1]. The lmt surface s obtaned by repeated refnements of the control curves, and has the form (S a P s )(t), s [0, 1], t R

22 22 Nra Dyn The qualty of the generated surface depends on the qualty of the parametrzaton of the ntal curves, as a reasonable prametrzaton for each set of refned curves at each refnement level: {S k a P s, s [0, 1]} k = 0, 1, 2,... In the nonlnear scheme the control curves at each refnement level are parametrzed, takng nto account ther geometry, and S a s appled to all control polygons generated by ponts on the curves correspondng to the same parameter value. The parametrzaton of the curves at refnement level k, {C k}n k =0 s n terms of a vector of correspondences τ = (τ 0,...,τ nk 1) between pars of consecutve curves. A correspondence between two curves s a one-toone and onto, contnuous map from the ponts of one curve to the ponts of the other. Thus τ determnes a parametrzaton of the curves n terms of the ponts of C0 k, n the sense that all the ponts P 0,..., P nk, wth P 0 C0 k and P +1 = τ (P ) C +1, correspond to the same parameter value. The convergence of the nonlnear scheme s proved for ntal curves contaned n a compact set n R 3. Ths condton s also satsfed by all control curves generated by the nonlnear scheme, due to the refnement rules of the curves. The correspondence used s a geometrcal correspondence τ defned by, τ = arg mn max{ τ(p) P 2, P C k τ T k (C k,ck +1 ) }, where T k (C k, Ck +1 ) s a set of allowed correspondences. Ths set depends on all the curves at refnement level k n a rather mld way. We ommt here the techncal detals. It s shown n Dyn, Elber, & Ita (2008) that f the ntal curves are admssble for subdvson, namely the sets T 0 (C 0, C0 +1 ) for = 0,...,n 0 1 are nonempty, then the sets T k (C k, Ck +1 ) for = 0,...,n k 1 are also nonempty for all k > 0. Wth the above notaton, the refnement step at refnement level k can be wrtten as: for = 0,...,n k 1, () compute τ () for each P C k, defne Q (P) = 3 4 P τ (P), R (P) = 1 4 P τ (P)

23 23 () defne two refned curves n k+1 = 2n k 1 C k+1 2 = {Q (P), P C }, C k = {R (P), P C } For the convergence proof we need an analoguous noton to the control polygon n the case of schemes refnng control ponts. Ths noton s the control pecewse-ruled-surface, defned at refnement level k by PR k = n k 1 =0 {P = λp + (1 λ)τ (P ), P C k, λ [0, 1]} Fg Geometrc refnement of curves. From left to rght: ntal curves, the refned curves after two and three refnement steps. It s shown n Dyn, Elber & Ita (2008) that f the ntal curves are smple, nonntersectng, and admssble for subdvson, then the sequence of control pecewse-ruled-surfaces {PR k } k 0 s well defned, and converges n the Hausdorff metrc to a set n R 3. Also t s shown there, that f the ntal curves are sampled densly enough from a smooth surface, then the lmt of the scheme approxmates the surface. From the computatonal pont of vew, the refnements are executed only a small number of tmes (at most 5), so the lmt s represented by the surface PR k wth 3 k 5. All computatons are done dscretely. The curves are sampled at a fnte number of ponts, and τ s computed by dynamcal programmng. An example demonstratng the performance of ths scheme on an ntal set of curves, sampled from a smooth surface, s gven n Fgure 1.5. References C. deboor (2001), A Practcal Gude to Splnes, Sprnger Verlag. A.S. Cavaretta, W. Dahmen, C.A. Mchell (1991), Statonary Subdvson, n Memors of the Amercan Mathematcal Socety 453.

24 24 Nra Dyn G.M. Chakn (1974), An algorthm for hgh speed curve generaton, Computer Graphcs and Image Processng 3, E. Cohen, T. Lyche & R.F. Resenfeld (1980), Dscrete b-splnes and subdvson technques, Computer Graphcs and Image Processng 14, I. Daubeches (1992), Ten Lectures on Wavelets, SIAM Publcatons. I. Daubeches, J.C. Lagaras (1992), Two-scale dfference equatons II, local regularty, nfnte products of matrces and fractals, SIAM J. Math. Anal. 23, G. Deslaurers, S. Dubuc (1989), Symmetrc teratve nterpolaton, Constructve Approxmaton 5, N. Dyn (1992), Subdvson schemes n Computer-Aded Geometrc Desgn, n Advances n Numercal Analyss - Volume II, Wavelets, Subdvson Algorthms and Radal Bass Functons, W. Lght (ed.), Clarendon Press, Oxford, N. Dyn (2005), Three famles of nonlnear subdvson schemes, n Multvarate Approxmaton and Interpolaton, K. Jetter, D. Buhmann, W. Haussmann, R. Schaback, J. Stöckler (eds.), Elsever, N. Dyn, G. Elber, U. Ita (2008), A subdvson scheme generatng surfaces by repeated refnements of curves, work n progress. N. Dyn, M.S. Floater, K. Hormann (2008), Four-pont curve subdvson based on terated chordal and centrpetal parametrzatons, preprnt. N. Dyn, J.A. Gregory. D. Levn (1987), A four-pont nterpolatory subdvson scheme for curve desgn, Computer Aded Geometrc Desgn 4, N. Dyn, J.A. Gregory. D. Levn (1991), Analyss of unform bnary subdvson schemes for curve desgn, Constructve Approxmaton 7, N. Dyn, D. Levn (1990), Interpolatng subdvson schemes for the generaton of curves and surfaces, n Multvarate Approxmaton and Interpolaton, W. Haussmann, K. Jetter (eds.), Brkhäuser Verlag, N. Dyn, D. Levn (1995), Analyss of asymptotcally equvalent bnary subdvson schemes, J. Mathematcal Analyss and Applcatons 193, N. Dyn, D. Levn (2002), Subdvson schemes n geometrc modellng, Acta Numerca, P. Grohs, J. Wallner (2008), Log-exponental analogues of unvarate subdvson schemes n Le groups and ther smoothness propertes, n Approxmaton Theory XII, M. Neamtu, L.L. Schumaker (eds.), Nashboro Press, J. Hechler, B. M 0ßner, U. Ref (2008), C 1 -contnuty of the generalzed fourpont scheme, preprnt. J. Lane, R. Resenfeld (1980), A theoretcal development for the computer generaton and dsplay of pecewse polynomal surfaces, IEEE Transactons on Pattern Analyss and Machne Intellgence 2, D. Levn (1999), Usng Laurent plynomal representaton for the analyss of the nonunform bnary subdvson schemes, Advances n Computatonal Mathematcs 11, M. Marnov, N. Dyn, D. Levn (2005), Geometrcally controlled 4-pont nterpolatory schemes, n Advances n Multresoluton for Geometrc Modellng, N. A. Dodgson, M.S. Floater, M. A. Sabn (eds.), Sprnger-Verlag, H. Prautzsch, W. Boehm, M. Paluszny (2002), Bézer and B-splne Technques, Sprnger. G. de Rahm (1956), Sur une courbe plane, J. de Math. Pures & Appl. 35,

25 I.U. Rahman, I. Dror, V.C. Stodden, D.L. Donoho, P. Schröder (2005), Multscale representatons for manfold-valued data, Multscale Modelng and Smulaton 4, J. Wallner (2006), Smoothness analyss of subdvson schemes by proxmty, Constructve Approxmaton 24, J. Wallner, N. Dyn (2005), Convergence and C 1 analyss of subdvson schemes on manfolds by proxmty, Computer Aded Geometrc Desgn 22, J. Wallner, E. Nava Yazdan, P. Grohs (2007), Smoothness propertes of Le group subdvson schemes, Multscale modelng and Smulaton 6, G. Xe, T.P.-Y. Yu (2007), Smoothness equvalence propertes of manfolddata subdvson schemes based on the proecton approach, SIAM ournal on Numercal Analyss 45, G. Xe, T.P.-Y. Yu (2008), Smoothness equvalence propertes of general manfold-valued data subdvson schemes, Multscale Modelng and Smulaton, to appear G. Xe, T.P.-Y. Yu (2008a), Approxmaton order equvalence propertes of manfold-valued data subdvson schemes preprnt. 25