Subdivision schemes with non-negative masks ... - Semantic Scholar

Baltzer Journals

March 31, 1996

Subdivision schemes with non-negative masks converge always - unless they obviously cannot? Avraham A. Melkman Department of Mathematics and Computer Science Ben Gurion University of the Negev, Beer Sheva, Israel, 84105

E-mail: [email protected]

It is conjectured that any non-negative stationary univariate subdivision scheme converges unless its mask has some obviously bad properties. In support of this conjecture several particular cases are proven, which subsume previously known results. It is conjectured further that the associated re nable function is positive on the interior of its interval of support, unless the subdivision scheme is interpolatory. Subject classi cation: AMS(MOS) 65D17, 41A15, 39A10. Keywords: subdivision, convergence, curve, functional equation.

1 Introduction Given an initial sequence of data values, v 0 = fvi0g, a stationary (binary) subdivision scheme with mask fa0; : : :; an g recursively de nes new sequences of values, v k , by applying the rule X (1) vik = ai?2j vjk?1 : j 2ZZ

This scheme is said to converge if for each v 2 `1 (ZZ ) there exists a continuous function fv such that i ) ? v k j = 0; lim sup j f ( v i k!1 2k i2ZZ

and fv 6 0 for some v . As is well known, a necessary condition for convergence is that X a = 1; i = 0; 1: (2) i?2j j 2ZZ

Henceforth it will be assumed implicitly that this condition is satis ed. The monograph by Cavaretta, Dahmen and Micchelli [2] is devoted to the anal-

A.A. Melkman /Non-negative subdivision converges always?

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ysis of general subdivision schemes, as is the review of Dyn [7]. Here we focus on subdivision schemes with non-negative coecients, a property possessed by many practical schemes in geometric modeling, [4] [6] [10]. This class of schemes has several remarkable properties. For instance, it is shown in [2] that such schemes always converge weakly. Moreover, the strong convergence of such schemes, or lack thereof, does not depend on the actual values of the mask coecients but rather on the support of the mask I = fi : ai > 0g; cf. Theorem .1. Consequently the question was raised in [2] to identify those I such that given any positive mask supported on I the corresponding subdivision scheme converges. Partial answers were given by Micchelli and Prautzsch [12], who showed that convergence is assured if I = f0; 1; : : :; ng, by Gonsor [9] who relaxed this condition to f0; 1; n ? 1; ng I , and by Wang (private communication), who relaxed this condition further to I f0; 1; 2pg or I f0; 2p ? 1; 2pg for some p > 0. Here we state the following

Conjecture 1

A non-negative subdivision scheme whose mask is supported on I , f0; ng I f0; 1; : : :; ng, converges if and only if both of the following hold: (a) either n is even and I contains at least one odd integer, or else I contains at least two odd and two even integers (including 0); (b) the greatest common divisor of the elements of I is 1.

Remark 1

In view of equality (2) condition (a) can be stated succinctly as the requirement that a0 ; an < 1. The necessity of this condition was also proven by Wang [14].

In support of this conjecture we show in section 3 that its conditions (a) and (b) are on the one hand necessary for convergence, and on the other hand satis ed in a number of cases in which convergence can be established. To cite but one example which subsumes Gonsor's condition, Theorem .7 proves convergence whenever I f0; p; q; p + q g with p and q relatively prime. We base our analysis on the following necessary and sucient condition for convergence of a subdivision scheme, proven in section 2. Its statement uses the standard notation for algebraic sums of sets, B + C = fb + c : b 2 B; c 2 C g, and 2B = f2b : b 2 B g.

Theorem .1

Given a non-negative subdivision scheme whose mask is supported on I de ne IN := I + 2I + + 2N ?1I: (3)


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Then the subdivision scheme converges if and only if there is an N such that for each i there exists a j satisfying i + ` ? 2N j 2 IN ; ` = 0; ; n ? 1:

With this theorem in mind let us make the following de nition.

De nition .2

A set of integers I fm; m + 1; : : :; m + ng has property P if there is an N such that for each i there exists a j satisfying i + ` ? 2N j 2 IN ; ` = 0; ; n ? 1: (4)

In these terms, then, Conjecture 1 can be rephrased to state that its conditions (a) and (b) are satis ed if and only if the set I has property P . It is this form of the conjecture that is examined in section 3. Associated with a convergent subdivision scheme is a re nable function which is the unique compactly supported solution of the functional equation X '(x) = ai'(2x ? i): i2ZZ

In fact, ' is the function obtained by subdivision from the initial data vi0 = 0;i . Thus ' is non-negative whenever the mask is non-negative. Micchelli and Pinkus [11] proved the conjecture of Micchelli and Prautzsch [12] that if a non-negative subdivision scheme has a mask with support I = f0; 1; : : :; ng then its associated re nable function is positive on its support, (0; n). We conjecture the following. Conjecture 2 Whenever a non-negative subdivision scheme is convergent its associated re nable function is positive on its support, (0; n), unless the scheme is interpolatory and n > 2. By interpolatory we mean that in the support of the mask there is a single integer that is even (or odd). For, if ` is that single integer we have by equality (2) that a` = 1 and hence v2ki+` = vik+?`1 . Any zero values in the initial data sequence will therefore be reproduced at each stage (at shifted parameter locations). Hence, if n > 2 and the scheme is interpolatory its re nable function, which is obtained by subdivision from the initial data f0;i g, must have zeros within its support. This shows that the condition of Conjecture 2 is indeed necessary. For support of the conjecture in the other direction, observe that if IN fa; : : :; n(2N ? 1) ? bg for some xed a; b and all N N0, then the method of proof of [11] is easily extended to show that the re nable function is indeed positive on (0; n). We show in section 3 that this is the case if n = p + q and I f0; p; q; p + q g with p and q relatively prime. Although we believe a similar statement is true in general we have been unable to prove it.


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2 A necessary and sucient condition for convergence This section is devoted to the proof of

Theorem .1

A non-negative subdivision scheme whose mask is supported on I converges if and only if there is an N such that for each i there exists a j satisfying

i + ` ? 2N j 2 IN ; ` = 0; ; n ? 1: The proof uses a formulation of subdivision, equivalent to equation (1), in terms of multiplication of the data vector v k by the bi-in nite matrix associated with the subdivision scheme,

A = fai?2j gi;j2ZZ : Thus v m = Am v 0 . A is a row-stochastic matrix, meaning its entries are nonnegative and its rows sum to one (because of equation (2)). Moreover, it is possible to determine precisely which entries of A are positive.

Lemma .3

Let A be the matrix associated with a non-negative subdivision scheme whose mask is supported on I . Then (Am )i;j > 0 if and only if i ? 2m j 2 Im .

Proof

The lemma is implicitly part of the proof of Buhman and Micchelli's [1] Theorem 2.3. The statement isPclearly true for m = 1. Proceeding by induction on m consider (Am+1 )i;j = k ai?2k (Am )k;j . By the induction assumption the entry will be non-zero if and only if there exists a k such that i ? 2k 2 I and k ? 2m j 2 Im. If this holds then clearly i ? 2m+1 j 2 I + 2Im = Im+1 . Conversely, if i ? 2m+1 j 2 I + 2Im, that is, i ? 2m+1 j = i1 + 2i2 for some i1 2 I and i2 2 Im , then k := 2m j + i2 satis es i ? 2k = i1 2 I , and k ? 2m j 2 Im . Thus the proof of Theorem .1 reduces to the proof of the following \positive column" condition on a power of A.

Lemma .4

Let A be the matrix associated with a non-negative subdivision scheme whose mask is supported on I; f0; ng I f0; 1; : : :; ng. Then the subdivision scheme converges if and only if there is an N such that for each i there exists a j satisfying

(AN )i+`;j > 0; ` = 0; ; n ? 1:

(1)


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Proof

The suciency of the condition is also proven, in much greater generality, in Cavaretta and Melkman [3]. Here we base our proof on the equivalence of the above subdivision scheme and the matrix re nement scheme of Micchelli and Prautzsch [13], as pointed out by Micchelli and Prautzsch [12], see also Dyn, Gregory and Levin [8] and Cavaretta, Dahmen and Micchelli [2]. Denote by Wi the n n submatrix of A ending at row i and column ci = i div 2, where i div 2 and i mod 2 are de ned by i = 2(i div 2) + i mod 2, with 0 i mod 2 1. We note the following facts. 1. All non-zero entries of A in rows i ? n + 1 through i are contained in Wi . 2. Wi = Wi mod2 . 3. All non-zero entries of Am in rows r ? n + 1 through r are contained in the n n submatrix Wrm formed by columns r1 ? n + 1 through r1, where r1 is de ned by ri?1 = ri div 2 ; i = 2; : : :; m; rm = r: (2) Indeed, it follows from fact 1 that

Wrm = Wr Wrm?1 Wr1 :

(3)

It is proven in Cavaretta, Dahmen and Micchelli [2] that the subdivision scheme converges if and only if the matrix re nement scheme based on W0 and W1 converges, while Micchelli and Prautzsch [13] prove that if W0 and W1 are stochastic, as is the case here, then the matrix re nement scheme converges if and only if there is an N such that any product

W"N W"1 ; "i 2 f0; 1g;

(4)

has a column with all entries positive. In the terminology of Daubechies and Lagarias [5], fW0; W1g forms a LCP-set. To complete the proof of the lemma it remains merely to observe that this last condition (4) is equivalent to the requirement of the lemma, that WrN have a positive column for all r. This is an immediate consequence of the fact that the set of matrices fWrN gr2ZZ is identical with the set of matrices given by products of the form (4). Indeed, given r de ne "k = rk mod 2, so that by fact 2 and equation (3), WrN = W"N W"1 : (5) P And, conversely, given "1 ; : : :; "N consider WrN with r de ned by r = iN=0?1 "N ?i 2i . P k ? 1 Then rk = i=0 "k?i 2i and rk mod 2 = "k ; 1 k N , so that equation (5) again holds.


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3 Which sets have property P ? In view of Theorem .1 we can reformulate Conjecture 1 as Conjecture 10 A set of integers I , f0; ng I f0; 1; : : :; ng, has property P if and only if both of the following hold: (a) either n is even and I contains at least one odd integer, or else I contains at least two odd and two even integers (including 0); (b) the greatest common divisor of the elements of I is 1. In support of the conjecture we prove rst that its conditions are indeed necessary, and then that they are also sucient in several restricted cases. We will use the following, easily ascertained, facts.

Lemma .5 P 1. IN = f iN=0?1 ci 2i : ci 2 I g:

2. If I has property P then so do ?I and f`g + I , for arbitrary `. 3. I does not have property P if, for each N , IN does not contain any integer of the form 2N k; k > 0. 4. I has property P if there is an N such that IN contains a sequence of consecutive integers of length 2N + m ? 1;

Theorem .6

The conditions of Conjecture 10 are necessary.

Proof

If gcd(I ) > 1 then it is seen from Lemma .5(1) that IN does not contain any successive integers, for any N . To prove the necessity of (a) we need consider only odd n. Suppose rst that I contains only odd positive integers. We shall prove by induction on N that then IN does not contain any integer of the form k2N ; k = 1; 2; , and hence by Lemma .5 (3) that I does not have property P . This is clearly true for N = 1. Suppose it is true for N but not for N + 1; using Lemma .5 (1) this means that for some k > 0 N X c 2i = k2N +1 :

But then

i=0

i

X c 2i = 2N (2k ? c

N ?1 i=0

i

N ):


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Since k > 0 and cN is either 0 or odd, 2k ? cN > 0, which contradicts the induction hypothesis. The case that I contains only even integers, in addition to the odd n, follows from Lemma .5 (2).

Theorem .7

I has property P if I f0; p; q; p + qg, with p < q relatively prime. Speci cally, for N log2 q , IN contains the integer interval [(p ? 1)(q ? 1); (p + q )(2N ? 1) ? (p ? 1)(q ? 1)]:

Proof

Observe that f0; p; q; p + q g = pf0; 1g + q f0; 1g. Hence IN pf0; 1; : : :; 2N ? 1g + qf0; 1; : : :; 2N ? 1g: It suces therefore to show that any ` 2 [(p?1)(q ?1); (p+q )(2N ?1)?(p?1)(q ?1)] can be represented as ap + bq with 0 a; b 2N ? 1. As a rst step we prove that for each ` 2 (p ? 1)(q ? 1) + [0; q ? 2] there exist a` ; b` such that a` p + b` q = ` and 0 a` q ? 2; 0 b` p ? 1. Since p and q are relatively prime f(ip) mod q : 0 i q ? 1g = f0; : : :q ? 1g. Hence, given ` 2 (p ? 1)(q ? 1)+[0; q ? 1] let a` be the unique integer, 0 a` q ? 1, such that (a` p) mod q = l mod q: Thus a` p = ((a` p) div q )q + ` mod q , and, with b` = ` div q ? (a` p) div q , ` = a` p + b`q: Note that for ` = (p ? 1)(q ? 1) + q ? 1 = p(q ? 1) we have a` = q ? 1 so that 0 a` q ? 2 for (p ? 1)(q ? 1) ` < p(q ? 1), while b` ` div q p ? 1. We show next that 0 b` . This is obviously true if a` p < (p ? 1)(q ? 1), since then b`q = ` ? a` p > 0. And if a` p 2 (p ? 1)(q ? 1) + [0; q ? 2] then it is the unique integer j in this interval with j mod q = ` mod q , i.e., a` p = ` and b` = 0. Since (p?1)(q ?1)+q ?1 = p(q ?1) is trivially representable as ap+bq it follows that ` can be written as ap + bq with 0 a 2N ? 1; 0 b 2N ? 1, whenever ` 2 ip + jq +(p ? 1)(q ? 1)+[0; q ? 1] for some i; j , 0 i 2N ? 1 ? (q ? 1); 0 j 2N ? 1 ? (p ? 1), i.e., ` 2 [(p ? 1)(q ? 1); (2N ? 1)(p + q ) ? q (p ? 1)]. Furthermore, ` can be written this way also if ` 2 (2N ? 1 ? (q ? 2))p + (2N ? 1 ? (p ? 1))q + (p ? 1)(q ? 1) + [0; q ? 2], completing the proof.

Theorem .8

Suppose I satis es the conditions of Conjecture 10 . Then I has property P in both of the following cases:


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1. I contains two successive integers; 2. I contains two odd integers and two successive even integers, or two even integers and two successive odd integers.

Proof

The proofs repeatedly use the observation that if a 2= IN then 21 (a ? b) 2= IN ?1 for any b 2 I , because IN = I + 2IN ?1. 1. Let ` and ` + 1 be the successive integers. By Lemma .5 (2) it is sucient to prove that I^ has property P , where I^ = I ? f`g if ` is even and ` m ? 2, and otherwise I^ = f` + 1g ? I . In the rst case, if ` = 2q > 0 we have I^ f?2q; 0; 1; n ? 2qg; and in the latter case, if ` = 2p ? 1 n ? 2 we have I^ f 0; 1; 2pg; nally, if ` = 0 or ` = n ? 1 we have I^ f 0; 1; 2pg, because the conditions of conjecture 10 hold. In summary, we distinguish between two cases: either I^ f0; 1; 2pg for some p > 0, or I^ f?2q; 0; 1; 2r +1g for some r q > 0. In both cases I^N f0; 1; : : : 2N ? 1g because I^ f0; 1g. The crux of the proof lies in establishing, under the stated conditions, that 2M 2 I^M for some M 1. For, from I^N = I^N ?M + 2N ?M I^M ; it then follows that I^N f0; 1; : : :; 2N ?M ? 1g + f2N g = f2N ; 2N + 1; : : :; 2N + 2N ?M ? 1g: Therefore I^N f0; 1; : : :; 2N + 2N ?M ? 1g: The theorem then follows from Lemma .5 (4) upon choosing N large enough so that 2N ?M n. case(a) I^ f0; 1; 2pg. Let M be such that 2M 2p. Then 2M 2 I^M . For supposing to the contrary that 2M 2= I^M it is found that 2M ?1 ? p 2= I^M ?1, contradicting I^M ?1 f0; 1; : : :; 2M ?1 ? 1g. case(b) I^ f?2q; 0; 1; 2r + 1g for some r q > 0. Let q = 2k (2s + 1) and let M be such that 2M ?k?2 r ? s. Then 2M 2 I^M . For otherwise 2M ?1+q 2= I^M ?1, and thus also 2?k (2M ?1+q ) 2= I^M ?1?k . Since the latter is an odd integer we conclude that 2M ?k?2 + s ? r 2= I^M ?k?2 , in contradiction to 0 2M ?k?2 + s ? r 2M ?k?2 ? 1. 2. By considering either I^ = I ?f`g or I^ = f` + 2g? I , it suces to prove that I^ has property P for I^ f0; 2; 2q + 1; 2p + 1g, with p > jqj 0. We will show that I^N contains an integer interval of length at least 2N +2N ?c for some xed c, which implies that I^ has property P by Lemma .5 (4).


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Speci cally, let p ? q = 2k (2s + 1), and let ` be any integer in the range (2p + 1)(2k+1 ? 1) ` 2N +1 ? 4 ? j2q + 1j. If ` is even then certainly ` 2 I^N , because I^N contains all even non-negative integers up to 2N +1 ? 2. Suppose then that ` is odd and de ne `1 = 21 (` ? 2p ? 1); m1 = 21 (` ? 2q ? 1), and `j = 21 (`j?1 ? 2p ? 1); mj = 12 (mj?1 ? 2p ? 1); 2 j k + 1: Then 0 (2p + 1)(2k+1?j ? 1) `j < mj 2N +1?j ? 2; 1 j k + 1; and mj ?`j = 21?j (p?q ). Now, since ` is odd, both `1 and m1 are integers not in I^N ?1, and hence both must be odd. We deduce similarly that both `k+1 and mk+1 are integers but not in I^N ?k?1 , and hence odd. This, however, is a contradiction since their dierence mk+1 ? `k+1 = 2s + 1 is an odd integer. Unfortunately, the method of proof of this theorem is not strong enough to establish the validity of Conjecture 2. Nor is the method applicable to the case of an interpolatory scheme, one in which n is even and I contains exactly one odd integer, say 2p + 1, except in the cases within the purview of this theorem, 2p + 1 = 1 or n ? 1. The reason is that IN cannot possibly contain a single interval of length at least 2N + m ? 1. To see this, let 2p + 1 be the odd integer and consider I^ := f2p + 1g ? I . Except for 0, I^ contains only odd integers. By the method of proof of Theorem .6 it can then be shown that I^N does not contain any integer of the form k2N , with k any non-zero integer, nor any of the integers 12 2N . Nevertheless, numerical experiments point to the conclusion that in this case too I has property P . For example, for I = f0; 3; 8g and N = 7 the following intervals were found: [30; 68]; [62; 140], and [126,164] (all taken mod 27).

Acknowledgements It is a pleasure to acknowledge enlightening discussions with M. Neamtu, who in uenced the proof of Theorem .8, and D. Berend, who contributed part 2 of that theorem. Y. Wang graciously provided us with his notes on this problem.

References [1] M.D. Buhmann and C. A. Micchelli. Using two-slanted matrices for subdivision, Proc. London Math. Soc., 69 (1994) 428{448.


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[2] A.S. Cavaretta, W. Dahmen, and C.A. Micchelli, Stationary Subdivision, volume 453 of Memoirs, American Mathematical Society, 1991. [3] A.S. Cavaretta and A.A. Melkman, Parametrization of subdivision schemes, in preparation, 1995. [4] G.M. Chaikin, An algorithm for high speed curve generation, Computer Graphics and Image Processing, 3 (1974) 346{349. [5] I. Daubechies and J. Lagarias, Sets of matrices all in nite products of which converge, Linear Algebra Appl., 161 (1992) 227{263. [6] G. de Rham, Sur une courbe plane, J. Mathem. pures et appl., 39 (1956) 25{42. [7] N. Dyn, Subdivision schemes in computer-aided geometric design, in Advances in Numerical Analysis II, W. Light, ed., Clarendon Press, Oxford, 1991, pp.36{104. [8] N. Dyn, J.A. Gregory, and D. Levin, Analysis of linear binary subdivision schemes for curve design, Constr. Approx., 7 (1991) 127{147. [9] D.E. Gonsor, Subdivision algorithms with nonnegative masks generally converge, Adv. Comp. Math., 1 (1993) 215{221. [10] J.M. Lane and R.F. Riesenfeld, A theoretical development for the computer generation of piecewise polynomial surfaces, IEEE Trans. Pattern Anal. Mach. Intell., 2 (1980) 35{46. [11] C.A. Micchelli and A. Pinkus, Descartes systems from corner cutting, Constr. Approx., 7 (1991) 161{194. [12] C.A. Micchelli and H. Prautzsch, Re nement and subdivision for spaces of integer translates of compactly supported functions, in Numerical Analysis, D.F. Griths and G.A. Watson eds., Longman, 1987, pp. 192{222. [13] C.A. Micchelli and H. Prautzsch, Uniform re nement of curves, Linear Algebra Appl., 114/115 (1989) 841{870. [14] Y. Wang, Two-scale dilation equations and the cascade algorithm, Random and Computational Dynamics, to appear.