350 Nra Dyn and Elza Farkh 1. Introducton. The nterest n developng subdvson schemes for compact sets s motvated by the problem of the reconstructon of

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1 Serdca Math. J. 28 (2002), SPLINE SUBDIVISION SCHEMES FOR COMPACT SETS. A SURVEY Nra Dyn and Elza Farkh Communcated by P.P.Petrushev Dedcated to the memory of our colleague Vasl Popov January 14, 1942 { May 31, 1990 Abstract. Attempts at extendng splne subdvson schemes to operate on compact sets are revewed. The am s to develop a procedure for approxmatng a set-valued functon wth compact mages from a nte set of ts samples. Ths s motvated by the problem of reconstructng a 3D object from a nte set of ts parallel cross sectons. The rst attempt s lmted to the case of convex sets, where the Mnkowsk sum of sets s successfully appled to replace addton of scalars. Snce for nonconvex sets the Mnkowsk sum s too bg and there s no approxmaton result as n the case of convex sets, a bnary operaton, called metrc average, s used nstead. Wth the metrc average, splne subdvson schemes consttute approxmatng operators for set-valued functons whch are Lpschtz contnuous n the Hausdor metrc. Yet ths result s not completely satsfactory, snce 3D objects are not contnuous n the Hausdor metrc near ponts of change of topology, and a specal treatment near such ponts has yet to be desgned Mathematcs Subject Classcaton: 26E25, 28B20, 41A15, 41A36. Key words: compact sets, splne subdvson schemes, metrc average, Mnkowsk sum. * Partally supported by ISF-Center of Excellence, and by The Hermann Mnkowsk Center for Geometry at Tel Avv Unversty, Israel

2 350 Nra Dyn and Elza Farkh 1. Introducton. The nterest n developng subdvson schemes for compact sets s motvated by the problem of the reconstructon of 3D objects from a set of ther 2D parallel cross sectons, or the reconstructon of a 2D shape from a set of ts 1D parallel cross sectons. For a revew on methods for the reconstructon of 3D objects from a nte set of parallel cross sectons see [10]. In our approach every n-dmensonal body s regarded as a unvarate set-valued functon wth compact sets of dmenson n ; 1 as mages, determned by parallel cross sectons [9]. The set-valued functon s then approxmated from the gven samples (cross sectons). The approxmatng procedure we use s an extenson to compact sets of splne subdvson schemes. A splne subdvson scheme generates from data consstng of real values attached to the nteger ponts, a smooth functon. In case of data sampled from a smooth functon, the lmt functon, generated by such ascheme, approxmates the sampled functon, and has shape preservng propertes [4, 5, 11]. Here we consder splne subdvson schemes operatng on data consstng of compact sets. A splne subdvson scheme generates from such ntal data a sequence of set-valued functons, wth compact sets as mages. Ths sequence converges n the Hausdor metrc to a lmt set-valued functon. In the case of 2D sets, the lmt set valued functon, wth 2D sets as mages, descrbes a 3D object. For the case of ntal data consstng of convex compact sets, we ntroduced n [6] splne subdvson schemes, where the usual addton of numbers s replaced by Mnkowsk sums of sets. Then the splne subdvson schemes generate lmt set-valued functons wth convex compact mages whch can be expressed as lnear combnatons of nteger shfts of a B-splne, wth the ntal sets as coecents. The subdvson technques are used to conclude that these lmt \set-valued splne functons" have shape preservng propertes smlar to those of scalar splne functons, but wth shape propertes relevant to sequences of sets and to set-valued functons. In the case of nonconvex ntal sets t s shown n [7] that the lmt setvalued functon, generated by asplne subdvson scheme, usng the Mnkowsk sums, concdes wth the lmt set-valued functon, generated by the same subdvson scheme from the convex hulls of the ntal sets. Therefore, a set-valued functon generated n such away, has too bg mages to be a good approxmaton to the set-valued functon from whch the ntal nonconvex sets are sampled. To dene splne subdvson schemes for general compact sets, whch do not convexfy the ntal data,.e. preserve the non-convexty, the usual Mnkowsk average s replaced by a bnary operaton between two compact sets, the metrc average, ntroduced n [1] and appled wthn subdvson schemes n

3 Splne Subdvson Schemes for Compact Sets. A survey 351 [8]. As s shown n [8], splne subdvson schemes, based on the metrc average, converge n the Hausdor metrc. The lmt set-valued functon generated by such ascheme, from ntal data sampled at dstance h from a Lpschtz contnuous set-valued functon wth compact mages, approxmates to order O(h) the sampled functon. 2. Prelmnares. Frst we ntroduce some notatons. The collecton of all nonempty compact subsets of R n s denoted by K n, C n denotes the collecton of convex sets n K n, h s the nner product n R n, jxj s the Eucldean norm of x 2 R n, S n;1 s the unt sphere n R n,coa denotes the convex hull of the set A. The Hausdor dstance between two sets A and B n R n s dened by haus(a B) = maxf sup dst(x B) sup dst(y A) g x2a y2b where the Eucldean dstance from a pont x to a set A 2K n s dst(x A) = mnfjx ; yj : y 2 A g: The support functon (A ) :R n! R s dened for A 2K n as (A l) = max a2a hl a l 2 Rn : The set of all projectons of x on the set A s The set derence of A B 2K n s A (x) =f a 2 A : ja ; xj = dst(x A) g: A n B = f a : a 2 A a 62 B g: A lnear Mnkowsk combnaton of two sets A and B s A + B = f a + b : a 2 A b 2 B g for A B 2K n and 2 R. The Mnkowsk sum A + B corresponds to a lnear Mnkowsk combnaton wth = = 1. A Mnkowsk average (a Mnkowsk convex combnaton) of two sets s a lnear Mnkowsk combnaton wth non-negatve, summng up to 1. We denote by S the class of multfunctons (set-valued functons) of the form, (1) F (t) = NX =1 A f (t) where N s nte and A 2 C n. We say that F 2 S s C k f n (1) f 2 C k for =1 ::: N.

4 352 Nra Dyn and Elza Farkh The notons convergence, contnuty, Lpschtz contnuty for set-valued functons or for sets, are to be understood wth respect to the Hausdor metrc (dstance). Let us recall that K n s a complete metrc space wth respect to ths metrc. 3. Splne subdvson schemes for ponts n R d. A splne curve n R d of degree m s dened by (2) X C(t) = P 0 B m(t ; ) 2Z for each t 2 R where P 0 = fp 0 2 R d 2 Zg are the control ponts and B m () s a B-splne of degree m. Due to the compact support of B m, the treatment of the case n (2) apples also to curves dened by a nte set of control ponts. The curve n (2) s the lmt of a sequence of pecewse lnear curves, each nterpolatng the ponts generated by the splne subdvson scheme S m at a certan renement level accordng to the renement step, (3) P k+1 wth the splne weghts a [m] = 2 Z nf0 1 ::: m+1g: X = a [m] ;2j P j k 2 Z k =0 1 2 ::: j2z m +1 =2 m =0 1 ::: m+1 and a [m] = 0 for For m = 1, the above scheme has coecents a 0 = 1 2, a 1 = 1, a 2 = 1 2, and the renement step s: (4) P k+1 P k+1 2 = 1 2 P k P k ;1 (5) 2+1 = P k : In ths case the lmt s a lnear splne curve, nterpolatng the ntal ponts fp 0 : 2 Zg: A quadratc splne curve s obtaned as a lmt n case m = 2, wth the well-known scheme of Chakn. The coecents of ths scheme are: a 0 = 1 4, a 1 = 3 4, a 2 = 3 4, a 3 = 1, and the renement steps 4 (6) (7) P k+1 2 = 1 4 P k P k ;1 P k = 3 4 P k P k ;1 :

5 Splne Subdvson Schemes for Compact Sets. A survey 353 An mportant result (see e.g. [4], [5]) s that the scheme (3), startng from fp 0 : 2 Zg2l1, d converges to a functon f() 2 C(R) d P,.e. lm sup k!1 2Z jf(2;k ); P k j = 0 f and only f lm sup k!1 t2r jf(t) ; 2Z P k h(2k t ; )j =0,whereh() s the \hat functon" (8) h(t) =n 1 ;jtj for jtj 1 0 otherwse. The lmt functon f(t) s denoted by S 1 m P Extenson to convex compact sets. The case of convex compact sets s nvestgated n [6]. We assume that the ntal data ff 0, 2 Zg are convex compact sets. Then the addton operaton n (3) s replaced by the Mnkowsk sum of sets, and the multplcaton of a set by a scalar s dened as (9) The renement step becomes (10) F k+1 A = f a : a 2 A g 2 R: X = a [m] ;2j F j k 2 Z k =0 1 2 ::: j2z We note that convex compact sets are generated at each step of (10), f F 0, 2 Z are compact and convex. It s shown n [6] that the set-valued splne functon X (11) F 1 m (t) = F 0 B m(t ; ) for each t 2 R 2Z s the unform lmt n the Hausdor metrc of the subdvson scheme, lm sup k!1 2Z haus(f 1 m (2 ;k ) F k )=0 P or equvalently, that lm sup k!1 t2r haus F 1 m (t) 2Z F k h(2k t ; ) = 0, where h() s the hat functon dened n (8). The proofs n [6] are based on the support functons parametrzaton of convex compact sets. The lnear (and orderng) propertes of the support functons, reectng the correspondng propertes of the Mnkowsk operatons on convex sets, allow to reduce the subdvson process on convex compact sets to subdvson on the support functons, and to apply known results on subdvson of scalar functons. An easy way (suggested bydavd Levn) to see that F 1 m n (11) s the lmt of the splne subdvson scheme, s based on the assocatvty and dstrbutvty

6 354 Nra Dyn and Elza Farkh of the P Mnkowsk sum and the postve-scalar multplcaton of sets. F 0 = 2Z F 0 [] wth Wrtng [] j = 1 for j = 0 j 6=. P we getf 1 m = F 0 S1 m []. Snce S 1 m [] = B m (;) (see e.g. [4, 5]), (11) follows. The splne subdvson schemes have the followng shape preservng propertes: 1. Monotoncty preservaton: If F 0 F 0 k +1 for all, then F F k +1 for all k, and F 1 m s monotone n the sense that F 1 m (t) F 1 m (t + h) forany t 2 R and h>0. 2. Convexty preservaton: If 2 F 0 +1 F 0 + F 0 k +2 for all, then 2F +1 + F k +2 for all k, and F 1 m s convex n the sense that ts graph s F k convex,.e., 2F 1 m (t + h) F 1 m (t)+f 1 m (t +2h) for all h t 2 R. As already mentoned, subdvson schemes for compact sets consttute a method for the approxmate reconstructon of 3D objects from ther 2D parallel crossectons, or, respectvely, of 2D shapes from ther 1D parallel crossectons. Thus the rate of approxmaton of these schemes s of mportance. Indeed, for contnuous set-valued functons we have an approxmaton result. If the set-valued functon G() has convex compact mages (t s not necessary that ts graph s convex), and s Lpschtz contnuous, that s, haus(g(t + t) G(t)) = O(t), and the ntal data for the splne subdvson scheme consst of samples of G, of the form F 0 = G(t), 2 Z, then haus(g(t) F 1 m (t)) = O(t): One can easly use the method of proof n [6] to show that f G() s only contnuous, then the rght-hand sde of the last estmate s O(!(G t t)), where!(g t t) s the modulus of contnuty of G dened n terms of the Hausdor dstance. The estmate haus(g(t) F 1 m (t)) = O((t) 2 ) s obtaned for a multfuncton G(t) whch has a support functon (G(t) l) wth second dervatve wth respect to t, unformly bounded n l 2 S n;1. Clearly, every multfuncton G from S\C 2 satses ths condton. It s well known that Mnkowsk averages wth equal weghts of a large number of nonconvex sets, tend as the number of sets grows, to the lmt of the averages wth equal wghts of the convex hulls of the sets. It turns out that for every splne subdvson scheme, snce a xed Mnkowsk convex combnaton of a

7 Splne Subdvson Schemes for Compact Sets. A survey 355 small number of sets s repeated an nnte number of tmes, the lmt set-valued functon equals to the lmt multfuncton obtaned by the same scheme from the convex hulls of the ntal sets [7]: X (12) F 1 m (t) = (cof 0 ) B m (t ; ) for each t 2 R 2Z The proof of (12) s based on the use of a measure of nonconvexty of a set, the so-called nner radus, whch s an upper bound for the Hausdor dstance between the set and ts convex hull. Two mportant ngredents are used n the proof. A Pythagorean type upper estmate for the nner radus of a Mnkowsk sum of compact sets by the nner rad of the summands, proved by Cassels [3], and the fact that the coecents of averagng n the renement step of the splne subdvson schemes (10) are non-negatve and sum up to 1. Wth these two ngredents t can be shown that the Hausdor dstance between the set F k and ts convex hull vanshes as k! 1, unformly n, as a geometrc progresson wth a rato less than 1. Therefore, wth the Mnkowsk averages, no approxmaton result can be expected for set-valued functons wth nonconvex mages. Ths falure of the Mnkowsk sum for nonconvex sets s n accordance wth the observaton that the Mnkowsk average of convex sets has propertes, whch do not hold for nonconvex sets. Let A B C 2C n,01. Then 1. A +(1; )A = A, 2 (0 1) 2. A +(1; )B = A +(1; )C =) B = C. These two propertes do not hold for nonconvex sets. Indeed, for a nonconvex set A 2K n, A +(1; )A A. Here s a smple example, showng that Mnkowsk averages for nonconvex sets are too bg. A = f0 1g, A n = 1 np A = 0 1n n 2n ::: 1. Moreover, n the =1 Hausdor metrc, lm n!1 A n = coa, demonstratng the convexcaton nature of Mnkowsk averagng processes. 5. The metrc average. A bnary operaton ntroduced n [1], and called n [8] \metrc average", has several propertes whch make t approprate for our purposes. Denton 5.1. Let A B 2K n and 0 t 1. The t-weghted metrc average of A and B s (13) A t B = ftfag +(1; t) B (a) :a2ag[ft A (b)+(1; t)fbg : b 2 Bg

8 356 Nra Dyn and Elza Farkh where the lnear combnatons above are n the Mnkowsk sense. The followng propertes of the metrc average are easy to observe [8]. Let A B C 2K n and 0 t 1, 0 s 1. Then 1. A 0 B = B A 1 B = A A t B = B 1;t A: 2. A t A = A. 3. A \ B A t B ta +(1; t)b co(a [ B). The metrc property of ths average, whch s essental for our applcatons and whch gave t ts name, s proved n [1]: 4. haus(a t B A s B)=jt ; sjhaus(a B): The metrc average of sets n R has several more propertes [2]. Let A B C 2K 1, D E 2C 1, t 2 [0 1] and let (A) denote the Lebesgue measure of the set A. Then D t E = td +(1; t)e, (A t B)=t(A)+(1; t)(b). (co(a t B) n (A t B)) = t(coa n A)+(1; t)(cob n B). A t B = A t C =) B = C. In the next example we have plotted the one-dmensonal sets A, B and the set C t = A t B n one pcture, gvng B at the y-coordnate 0, A at y=1, and C t at y= t for t = (see Fgure 1). The two sets are 4 A =[0 1] [ [5 6] [ [7:5 8] [ [9 10] [ [11:5 14] B =[1 4] [ [5 6:5] [ [14 16]: It follows from the denton of the metrc average that the metrc average of two sets produces a subset of the Mnkowsk average and also that the metrc average of a set wth tself s the set. Indeed, ths bnary operaton, beng smaller than the Mnkowsk average, does not convexfy repeated averagng processes. Snce t s dened as a bnary operaton between two sets, n order to use t n splne subdvson schemes we need another representaton of these schemes n terms of repeated bnary averagng. 6. Splne subdvson schemes wth metrc averages. Frst, we represent the splne subdvson schemes n terms of repeated bnary averages.

9 Splne Subdvson Schemes for Compact Sets. A survey 357 C = A C 3/4 C 1/2 C 1/4 C = B Fgure 1. The sets A, B and C t The renement step(3)can be obtaned by one step of renement of the lnear splne subdvson, followed by a sequence of bnary averages. The sequence of steps whch replaces (3) conssts of rst denng (14) P k = P k P k = 1 2 (P k + P+1) k and then denng for 1 j m ; 1 the ntermedate averages (15) where (16) P k+1 j = 1 k+1 j;1 (P + P k+1 j;1 +1 ) 2 I j 2 I j = The nal values at level k + 1 are P k+1 P k = P k+1 m;1 = P k+1 m;1 ; 1 2 Z j odd Z n Z j even: for m odd 2 Z for m even 2 Z: 2 Z For example, n case m = 2, one step of (14) followed by one step of (15) s equvalent to the renement step of the Chakn scheme. The above procedure s carred over to compact sets, wth the metrc average replacng the averagng operatons n (14) and (15) [8]. Frst renng wth metrc averages (17) F k = F k F k+1 0 = F k F k Z

10 358 Nra Dyn and Elza Farkh and then, for 1 j m ; 1, replacng the sequence ff k+1 j;1 : 2 Zg by metrc averages of pars of consecutve sets (18) F k+1 j The nal rened sets are F k+1 = F k+1 j;1 = F k+1 m;1 1 2 F k+1 j; I j : for m odd 2 Z F k+1 = F k+1 m;1 ; 1 for m even 2 Z: 2 The convergence of ths scheme follows from the metrc property of the metrc average. Denote d k = sup haus(f k Fk +1 ) k = 0 1 :::. Then dk dk. At the k-th stage of the subdvson, the set-valued functon F k (t) s constructed as follows: F k (t) =F k (t) F k +1 2;k t ( +1)2 ;k where (t) = ( +1); t2 k. It follows from the metrc property of the metrc average that sup haus(f k+1 (t) F k (t)) = O(2 ;k ) t therefore ff k (t)g k2z + s a Cauchy sequence n the complete metrc space K n. Thus the lmt of ths sequence exsts and s denoted by S 1 m F 0 (t). The approxmaton property below justes the reconstructon of objects from ther parallel cross sectons by a splne subdvson scheme whch uses metrc averages nstead of Mnkowsk averages. Let the unvarate multfuncton G(t) have compact mages and let t be Lpschtz contnuous. If F 0 = G(t) 2 Z, then sup haus(s 1 m F 0 (t) G(t)) = O(t): t An example [8] of a shell ncluded between two quarters of spheres s represented n Fgure 2. Ths body can be represented by the followng set-valued functon F (x), dened for 0 x 1: F (x) =f (y z) 2 R 2 j z 0 r(x) y 2 + z 2 0:2+r(x) g where r(x) =1; x 2. Gven the ntal crossectons F (0) F(h) F(2h) ::: F(1), we reconstruct ths shell by a metrc subdvson scheme of Chakn type, and obtan a sequence of pecewse lnear (n a metrc sense) set-valued functons ff k (t)g 1 k=0, wth F k (t) nterpolatng the sets generated at level k. The crossectons F 3 h 2 +0:25, = of F 3, obtaned after three subdvson teratons from the ntal sets as above wth h = 0:125, are presented n Fgure 3.

11 z z Splne Subdvson Schemes for Compact Sets. A survey y Fgure 2. A shell ncluded between two quarters of spheres x y x 1 Fg. 3. Four cross-sectons of the nal body The maxmal Hausdor dstance between these crossectons at the thrd teraton and the correspondng crossectons of the ntal object s 0:0122. Snce 2D shapes and 3D objects, when regarded as unvarate multfunctons, are usually dscontnuous n the Hausdor metrc at ponts of change of topology, the above approxmaton result does not hold near such ponts. Ths observaton calls for a specal treatment near ponts of change of topology, a subject whch s stll under nvestgaton.

12 360 Nra Dyn and Elza Farkh REFERENCES [1] Z. Artsten. Pecewse lnear approxmatons of set-valued maps. Journal of Approx. Theory 56 (1989), [2] R. Baer, N. Dyn, E. Farkh. Metrc averages of one dmensonal compact sets. In: Approxmaton theory X (Eds C. Chu, L. L. Schumaker and J. Stoeckler), Vanderblt Unv. Press, Nashvlle, TN, 2002, 9{22. [3] J. W. S. Cassels. Measures of the non-convexty of sets and the Shapley{ Folkman{Starr theorem. Math. Proc. Camb. Phl. Soc. 78 (1975), [4] A. S. Cavaretta, W. Dahmen, C. A. Mcchell. Statonary Subdvson, Memors of AMS, No. 453, [5] N. Dyn. Subdvson schemes n computer-aded geometrc desgn. In: Advances n Numercal Analyss, Vol. II, Wavelets, Subdvson Algorthms and Radal Bass Functons, (Ed. W. Lght), Clarendon Press, Oxford, 1992, 36{ 104. [6] N. Dyn, E. Farkh. Splne subdvson schemes for convex compact sets. Journal of Comput. Appl. Mathematcs 119 (2000), [7] N. Dyn, E. Farkh. Convexcaton rate n Mnkowsk averagng processes. Preprnt. [8] N. Dyn, E. Farkh. Splne subdvson schemes for compact sets wth metrc averages. In: Trends n Approxmaton Theory, (Eds K. Kopotun, T. Lyche and M. Neamtu), Vanderblt Unv. Press, Nashvlle, TN, 2001, 95{ 104. [9] D. Levn. Multdmensonal reconstructon by set-valued approxmatons. IMA Journal of Numercal Analyss 6 (1986), [10] L. L. Schumaker. Reconstructng 3D objects from cross-sectons. In: Computaton of Curves and Surfaces, (Eds W. Dahmen, M. Gasca and C. Mcchell) Nato ASI seres, Kluwer Academc Publshers, 1990, [11] I. Yad-Shalom. Geometrc propertes of curves and surfaces generated by translates of a sngle functon. Ph. D. thess, Tel-Avv Unversty, School of Mathematcal Scences Sackler Faculty of Exact Scences Tel-Avv Unversty Tel Avv, Israel e-mal: nradyn@post.tau.ac.l, elza@math.tau.ac.l Receved September 30, 2002