# A ternary interpolatory subdivision schemes originated from splines

July 11, S0219691311004249 2011 15:46 WSPC/S0219-6913 181-IJWMIP International Journal of Wavelets, Multiresolution and Information Processing Vol. 9, No. 4 (2011) 611–633 c World Scientific Publishing Company DOI: 10.1142/S0219691311004249 TERNARY INTERPOLATORY SUBDIVISION SCHEMES ORIGINATED FROM SPLINES AMIR Z. AVERBUCH∗ and VALERY A. ZHELUDEV† School of Computer Science, Tel Aviv University Tel Aviv 69978, Israel ∗[email protected] †[email protected] GARY B. FATAKHOV Department of Applied Mathematics School of Mathematical Sciences Tel Aviv University, Tel Aviv 69978, Israel fat [email protected] EDUARD H. YAKUBOV Department of Sciences Holon Academic Institute of Technology 52 Golomb Street, Holon 58102, Israel [email protected] Received 1 January 2010 Revised 3 June 2010 A generic technique for construction of ternary interpolatory subdivision schemes, which is based on polynomial and discrete splines, is presented. These schemes have rational symbols. The symbols are explicitly presented in the paper. This is accompanied by a detailed description of the design of the refinement masks and by algorithms that verify the convergence of these schemes. In addition, the smoothness of the limit functions is investigated. The ternary subdivision schemes, whose construction is based on continuous splines, become tools for fast computation of interpolatory splines of arbitrary order at triadic rational points. Keywords: Subdivision; spline; interpolation; recursive filtering. AMS Subject Classification: 65D17, 65D07, 93E11 1. Introduction Subdivision started as a tool for eﬃcient computation of spline functions. Now, it is an independent subject with many applications. It is being used for the development of new methods for curve and surface design, approximation, generation of wavelets and multiresolution analysis and also for solving some classes of functional 611 July 11, S0219691311004249 612 2011 15:46 WSPC/S0219-6913 181-IJWMIP A. Z. Averbuch et al. equations. Many research paper have been written over the years on various subdivision schemes including binary subdivision schemes. Ternary subdivision schemes were investigated in Refs. 12 and 13 where it was showed that there exists a family of three and four point ternary subdivision schemes, whose limit functions belong to C 1 and C 2 , respectively. They used ﬁnite reﬁnement masks and showed that the fundamental functions, which were generated by ternary subdivision schemes, have smaller support than their binary counterparts. For continuity analysis of the ternary schemes, they used the generating function formalism technique that was developed in Ref. 1. A generic technique for construction of diﬀerent interpolatory binary subdivision schemes, which are based on polynomial and discrete splines, was introduced in Ref. 3. These schemes have rational symbols and inﬁnite masks but they are competitive (regularity, speed of convergence, computational complexity) with schemes that have ﬁnite masks. Exponential decay of the basic limit functions of schemes with rational symbols is proved in Ref. 3. This property guarantees the convergence of such schemes on initial data of power growth. A similar approach resulted in construction of a diverse library of wavelet and frame transforms, which proved to be eﬃcient in signal and image processing applications, see Refs. 4–6. In the present paper, we investigate univariate Ternary Interpolatory Subdivision Schemes (TISS) that are derived from continuous and discrete interpolatory splines. Our analysis extends the technique that was developed in Refs. 1–3 for binary schemes. We present a detailed analysis that enables to verify the convergence and the smoothness of TISS. We also show how to derive reﬁnement masks from continuous and discrete splines. A fast algorithm for computation of interpolatory splines of arbitrary order at triadic rational points is described. The paper is organized as follows. Basic notation and fundamentals of subdivision schemes are given in Sec. 2. Section 3 presents the main results on TISS. Sections 4 and 5 show how to design reﬁnement masks and how to implement the corresponding TISS. Examples of spline-based TISS are given in Sec. 6. The Appendix describes how to evaluate the coeﬃcients of TISS with inﬁnite masks using the discrete Fourier transform. In addition, the scheme for verifying the convergence of TISS is also presented in the Appendix. 2. Notation and Fundamentals of Interpolatory Subdivision Schemes Interpolatory Subdivision Schemes (ISS) are reﬁnement rules, which iteratively reﬁne the data by inserting values that correspond to intermediate points. This is done by using linear combinations of values in the initial points, while the data in these initial points are retained. Non-interpolatory schemes also update the initial data in addition to the inserted values in intermediate points. Stationary schemes use the same insertion rule at each reﬁnement step. A scheme is called uniform if its insertion rule does not depend on the location in the data. To be more speciﬁc, a July 11, S0219691311004249 2011 15:46 WSPC/S0219-6913 181-IJWMIP Ternary Interpolatory Subdivision Schemes Originated from Splines 613 univariate stationary uniform subdivision scheme with ternary reﬁnement, denoted by Sa , consists of the following: The function {f k }, which is deﬁned on the grid Gk = {j/3k }j∈Z : f k (j/3k ) = fjk , is extended onto the grid Gk+1 by ﬁltering the array {fjk }j∈Z to become: fjk+1 = aj−3l flk . (2.1) l∈Z This is one reﬁnement step. The next reﬁnement step employs fjk+1 as an initial data. The array a = {ak }k∈Z is called a reﬁnement mask of a subdivision scheme S. We assume that the series k∈Z ak is absolutely convergent. A subdivision scheme S with the reﬁnement mask a is denoted by Sa . The z-transform of a mask a is deﬁned by a(z) = k∈Z ak z k . It is also called the symbol of Sa . Throughout the paper, we assume that z = e−iω . We work with inﬁnite masks and assume that there exist Laurent polynomials T (z) and P (z) such that the symbol is T (z) . ak z k = a(z) = P (z) k∈Z If Sa is interpolatory then a0 = 1 and a3k = 0 for all k ∈ Z, k = 0. The sequence of the values fjk at level k is represented by its z-transform Fk (z) that is formally deﬁned as Fk (z) = j∈Z fjk z j . Equation (2.1) implies that k Fk+1 (z) = a(z)Fk (z 3 ) ⇒ Fk+1 (z) = F1 (z 3 ) k−1 i a(z 3 ). (2.2) i=0 The TISS Eq. (2.1) can be split into: k+1 k f3i = fi k . f k+1 = a3j±1 fi−j 3i±1 j∈Z Deﬁnition 2.1. Given an initial data f 0 = {fj0 } ∈ 1 , j ∈ Z, denote by f k (t) the sequence of polygonal lines (second-order splines) that interpolate the data generated by Sa at the corresponding reﬁnement level {f k (3−k j) = fjk = (Sak f 0 )j }, j ∈ Z. If {f k (t)} converges uniformly at any ﬁnite interval to a continuous function f ∞ (t), as k → ∞, then we say that the subdivision scheme Sa converges on the initial data f 0 and f ∞ (t) is called its limit function. If Sa converges for any f 0 ∈ 1 then Sa is called the convergent TISS. 3. Convergence and Regularity of TISS with Inﬁnite Masks 3.1. Preliminary results We restrict the admissible initial data to the sequences f = {fj0 } ∈ 1 , j ∈ Z. The symbol a(z) = T (z)/P (z) is subject to the following requirements: P1: The Laurent polynomials P (z) and T (z) are symmetric about inversion: P (z −1 ) = P (z), T (z −1 ) = T (z). Thus, they are real on the unit circle |z| = 1. July 11, S0219691311004249 614 2011 15:46 WSPC/S0219-6913 181-IJWMIP A. Z. Averbuch et al. P2: The roots of the denominator P (z) are real, simple and do not lie on the unit circle |z| = 1. P3: The symbol a(z) is factorized as follows: a(z) = (1 + z −1 + z −2 )q(z), q(1) = 1. (3.1) In the sequel, we say that a subdivision scheme Sa belongs to the class P if its symbol a(z) possesses the properties P1–P3. The above properties imply that the coeﬃcients ak of the mask of the scheme Sa of the class P are symmetric about zero. If P1 and P2 hold then P (z) can be represented as follows: P (z) = r 1 (1 + γn z)(1 + γn z −1 ), γ n n=1 0 < |γ1 | < |γ2 | < · · · < |γr | = e −g (3.2) < 1, g > 0. Proposition 3.1. If the symbol of a scheme Sa is a(z) = T (z)/P (z) and Eq. (3.2) holds then the mask satisﬁes the inequality |aj | ≤ Ae −g|j| , where A is a positive constant. Proof. Assume that the degree t of T (z) is less than the degree p of P (z). If Eq. (3.2) holds then the symbol can be represented as follows: A− A+ n nz + 1 + γn z 1 + γn z −1 n=1 r ∞ ∞ A+ (−γn )j z j + zA− (−γn )j z −j = n n a(z) = r n=1 j=0 j=0 ∞ + j 1−j = aj z + a− , j z a+ j = r j A+ n (−γn ) , n=1 j=0 a− j = r j A− n (−γn ) , n=1 where j |a+ j | ≤ |γr | r n=1 −gj |A+ , n | ≤ Ae j |a− j | ≤ |γr | r −gj |A− . n | ≤ Ae (3.3) n=1 If t ≥ p then a polynomial of degree t−p is added to the expansion (3.3). Obviously, this addition does not aﬀect the decay of the mask a(j) as j tends to inﬁnity. Lemma 3.1. Let Sa be the subdivision scheme whose symbol is a(z) = T (z)/P (z) and the Laurent polynomial P (z) satisﬁes the properties P1 and P2. If Eq. (3.2) July 11, S0219691311004249 2011 15:46 WSPC/S0219-6913 181-IJWMIP Ternary Interpolatory Subdivision Schemes Originated from Splines 615 holds then for any ﬁnite initial data f 0 the following inequalities −k+1 |fjk | ≤ Ak e−gj3 (3.4) hold, where Ak are positive constants. Proof. The mask of the scheme Sa decays exponentially, i.e. |aj | ≤ Ae−gj . Due ∆ to Eq. (2.2), F1 (z) = a(z)F0 (z 3 ) = T1 (z)/P1 (z), where T1 (z) = T (z)F0 (z 3 ) and P1 (z) = P (z). Hence, the roots of P1 (z) are ρ1n = −γn , 1 ≤ n ≤ r, and, therefore, |fj1 | ≤ A1 e−gj . The next reﬁnement step produces the following z-transform: F2 (z) = a(z)F1 (z 3 ) = T2 (z) , P2 (z) P2 (z) = P (z)P (z 3 ). The roots of P2 (z) satisfy the inequality |ρ2n | ≤ A2 e−gj/3 . Then, Eq. (3.4) is derived by induction. 3 |γr | = e−g/3 . Hence, |fj2 | ≤ ∆ k Denote ∆fjk = fj+1 − fjk and the z-transform of the diﬀerence sequence ∆fjk by Qk (z). Then, Qk (z) = (z −1 − 1)Fk (z). Proposition 3.2. If Eq. (3.1) holds then Qk+1 (z) = q(z)Qk (z 3 ). (3.5) Proof. Qk+1 (z) = (z −1 − 1)Fk+1 (z) = (z −1 − 1)a(z)Fk (z 3 ) = (z −1 − 1)(z −2 + z −1 + 1)q(z)Fk (z 3 ) = q(z)(z −3 − 1)Fk (z 3 ) = q(z)Qk (z 3 ). Equation (3.5) implies that there exists a ternary subdivision scheme of diﬀerences Sq deﬁned by: qj−3l ∆flk . (3.6) Sq : ∆fjk+1 = l∈Z ∆ Denote by F (z; f ) the z-transform of a sequence f = {fj }. Then, the z-transform of ∆ ∆ the diﬀerence sequence ∆f = {∆fj } is F (z; ∆f ) = (z −1 −1)F (z; f ) and F (z; Sa f ) = a(z)F (z 3 ; f ). We get F (z; ∆Sa f ) = (z −1 − 1)F (z; Sa f ) = (z −1 − 1)a(z)F (z 3 ; f ) = (z −1 − 1)(z −2 + z −1 + 1)q(z)F (z 3 ; f ) = (z −3 − 1)q(z)F (z 3 ; f ) = q(z)(z −3 − 1)F (z 3 ; ∆f ) = F (z; Sq ∆f ). Thus, ∆(Sa f ) = Sq ∆f. July 11, S0219691311004249 616 2011 15:46 WSPC/S0219-6913 181-IJWMIP A. Z. Averbuch et al. Example 1. TISS originated from the second-order (ﬁrst degree) interpok+1 −k as the values at the points (i ± 1/3) latory splines. We deﬁne the values f3i±1 −k3 k of the piecewise linear spline, which interpolates the data fi on the grid 3 i . Then, k+1 k f3i = fi 1 k 2 k+1 f3i−1 = · fi−1 + · fik , 3 3 2 1 k+1 k f3i+1 = · fik + · fi+1 3 3 Fk+1 (z) = alin (z)Fk (z 3 ), ∆ alin (z) = (z 2 + z + 1)2 . 3z 2 (3.7) 3.2. Convergence of subdivision schemes ∆ Denote f k ∞ = maxi∈Z |fik |. Then, we have f k+1 ∞ ≤ Sa f ∞ , k ∆ where Sa = max |a3k |, k∈Z |a3k+1 |, k∈Z |a3k+2 | . k∈Z After L reﬁnement steps, the following inequality holds [L] k+L L k L ∆ f ∞ ≤ Sa f ∞ , where Sa = max a3L k+n , n k∈Z n = 0, . . . , 3L − 1, [L] [L] and ak is the mask of the operator Sa L−1 a(z), . . . , a(z 3 ). (3.8) whose symbol is a[L] (z) = Theorem 3.1. Let Sa be a TISS of Class P and Sq be the subdivision scheme of diﬀerences deﬁned by Eq. (3.6). The scheme Sa converges if for some L ∈ N SqL ∞ = µ < 1. (3.9) Remark. A subdivision scheme whose norm satisﬁes the inequality (3.9) is called contractive. Proof. Recall that f k (t) are the second-order splines that interpolate the subsequently reﬁned data f k (3−k i) = fik , i ∈ Z, where the initial data is {fi0 }. We prove that {f k (t)}k∈Z is a Cauchy sequence that is for a given ε > 0 there exists N1 ∈ N such that for all n, m > N1 : supt∈R |f n (t) − f m (t)| < ε. ∆ k+1 (t), where Dk+1 (t) = f k+1 (t)− f k (t). We can write f n (t)− f m (t) = n−1 k=m D k+1 k+1 k+1 k k = fik , g3i−1 = (fi−1 + 2fik )/3, g3i+1 = (2fik + fi+1 )/3. From Denote g3i 3 2 −2 Eq. (3.7) we have Gk+1 (z) = alin (z)Fk (z ), where alin (z) = z (z + z −1 + 1)2/3. July 11, S0219691311004249 2011 15:46 WSPC/S0219-6913 181-IJWMIP Ternary Interpolatory Subdivision Schemes Originated from Splines 617 The maximal absolute value of the piecewise linear function Dk+1 (t) at the interval [3−k i, 3−k (i + 1)] is reached at its breakpoints. Therefore, |Dk+1 (t)| ≤ max f k+1 − g k+1 , f k+1 − g k+1 , 3i−1 sup |D k+1 (t)| = sup |f t∈R k+1 3i−1 (t) − g k+1 3i+1 (t)| = f 3i+1 k+1 t∈R − g k+1 ∞ . (3.10) From Eq. (3.1), we have a(z) = (z −2 + z −1 + 1)q(z), q(1) = 1. Thus, q(z) − (z + z 1 + 1)/3 = (z −1 − 1)r(z), r(z) = n∈Z rn z n and we have 2 Fk+1 (z) − Gk+1 (z) = a(z)Fk (z 3 ) − alin (z)Fk (z 3 ) (z 2 + z 1 + 1) −2 −1 = (z + z + 1) q(z) − Fk (z 3 ) 3 = (z −2 + z −1 + 1)(z −1 − 1)r(z)Fk (z 3 ) = (z −3 − 1)r(z)Fk (z 3 ) = r(z)Qk (z 3 ) (3.11) k where Qk (z) = (∆f )(z). The rational function r(z) has the same denominator as a(z) and, by Proposition (3.1), the coeﬃcients |rn | ≤ Ce−g|n| , g > 0. Therefore, |r(z)| ≤ n∈Z |rn | = R < ∞. Equation (3.11) implies that ri−3j ∆fjk . (3.12) fik+1 − gik+1 = j∈Z By combining Eqs. (3.10) and (3.12) we get sup |f k+1 (t) − g k+1 (t)| = f k+1 − g k+1 ∞ t∈R ≤ k = ri−3j ∆fj j∈Z ∞ |rn | · ∆f ∞ = R · ∆f ∞ = R · Sqk ∆f 0 ∞ . k k n∈Z Using Eq. (3.9), we obtain n−1 n−1 sup |f n (t) − f m (t)| = sup [f k+1 (t) − f k (t)] = sup [f k+1 (t) − g k+1 (t)] t∈R t∈R t∈R k=m ≤ n−1 k=m ≤ n−1 k=m sup |f k+1 (t) − g k+1 (t)| ≤ R · t∈R n−1 Sqk ∆f 0 ∞ k=m k Rµ L · max |∆f i | ≤ A · η n , 0≤i<L k=m 1 where η = µ L < 1, A > 0, n > L. The proof of the next proposition is straightforward. Proposition 3.3. If TISS Sa converges on the initial data f 0 and f k = Sa f k−1 then lim ∆df k = 0. k→∞ (3.13) July 11, S0219691311004249 618 2011 15:46 WSPC/S0219-6913 181-IJWMIP A. Z. Averbuch et al. Remark. The key practical problem in the application of TISS algorithms with inﬁnite masks is the evaluation of the sums of the coeﬃcients in (3.9). We present in the Appendix a method to evaluate the coeﬃcients via the discrete Fourier transform and an algorithm for verifying the convergence of such subdivision schemes. 3.2.1. Basic limit function Deﬁnition 3.1. Let Sa be a convergent TISS. Assume that the initial data is the ∆ Kronecker delta f 0 = {δ(k)}k∈Z . Then, the limit function ϕa (t) = Sa∞ f 0 (t) is called the basic limit function (BLF). If the mask of the subdivision scheme is ﬁnite then its BLF exists and has a compact support. This is not the case for the schemes with inﬁnite masks. However, for the class of TISSs that we deal with, the BLFs exist and decay exponentially when their arguments grow. Theorem 3.2. Let Sa be a TISS of Class P and Sq be the subdivision scheme of diﬀerences deﬁned by Eq. (3.6). If for some L ∈ N, the inequality (3.9) holds, then there exists a continuous BLF ϕa (t) of the scheme Sa , which decays exponentially as t → ∞. Namely, if (3.2) holds then for any ε > 0 a constant Φε > 0 exists such that the following inequality |ϕa (t)| ≤ Φε e−(g−ε)|t| holds. A similar fact about the binary ISS has been established in Ref. 3. The proof of the statement for the TISS almost literally coincides with the proof in the binary case provided in Ref. 3. Therefore, we omit it in the present paper. We also refer to Ref. 3 for the proof of the following fact. Corollary 3.1.3 Assume that TISS Sa satisﬁes the conditions of Theorem 3.2. Then, for any initial data f 0 = {fj0 }j∈Z , the limit function f ∞ (t) can be represented by the sum Sa∞ f 0 (t) = fj0 ϕa (t − j), (3.14) j where ϕa (t) is the BLF of the scheme Sa . 3.3. Smoothness of the limit functions In this section, we establish conditions for a TISS to produce a limit function that possesses some number of derivatives. Lemma 3.2. Assume that the support supp(ψ) of a function ψ ∈ C(R) is compact and the identity i∈Z ψ(x − i) = 1, x ∈ R, holds. If Sa is a convergent TISS then (S k f 0 )j ψ(3k t − j) = S ∞ f 0 (t). (3.15) lim k→∞ j∈Z July 11, S0219691311004249 2011 15:46 WSPC/S0219-6913 181-IJWMIP Ternary Interpolatory Subdivision Schemes Originated from Splines 619 Proof. We evaluate the diﬀerence ∆ k 0 k ∞ 0 ek (x) = (S f )j ψ(3 x − j) − S f (x) j∈Z = [(S k f 0 )j − S ∞ f 0 (x)]ψ(3k x − j) j∈Z |(S k f 0 )j − S ∞ f 0 (3−k j)||ψ(3k x − j)| ≤ j∈Z + |S ∞ f 0 (3−k j) − S ∞ f 0 (x)||ψ(3k x − j)| j∈Z = |S ∞ f 0 (3−k j) − S ∞ f 0 (x)||ψ(3k x − j)|, j∈Γ3k x where Γ3k x = {j : 3k x − j ∈ supp(ψ) ∧ j ∈ Z}. Due to compactness of supp(ψ), Γ3k x is a ﬁnite set. For j ∈ Γ3k x , x − 3−k j = y ∈ 3−k supp(ψ). Due to the uniform convergence of Sa at the interval Ωx,δ = {y : x − y∞ < δ} and due to the continuity of S ∞ f 0 (x), for any > 0, there exists N (, Ωx,δ ) such that ek (x) ≤ M ψ∞ , ∀ k > N (, Ωx,δ ). ∆ ∆ k We introduce the divided diﬀerences (df k )j = 3k ∆f k = 3k (fj+1 − fjk ), djf k = 3kj ∆jf k . It follows from Proposition 3.2 that if Sa is a TISS, whose symbol a(z) is factorized as a(z) = (1 + z −1 + z −2 )q(z), q(1) = 1, then the subdivision scheme Sa[1] is such that df k+1 = Sa[1] df k (3.16) has the symbol a[1] (z) = 3q(z) = 3a(z)/(z −2 + z −1 + 1). Denote ∆ a[j] (z) = 3j a(z)/(z −2 + z −1 + 1)j , ∆ q [j] (z) = a[j] (z)/(z −2 + z −1 + 1) = a[j+1] (z)/3, j = 0, 1 . . . . . (3.17) The function q [j] (z) is the symbol of the subdivision scheme of diﬀerences Sq[j] : [j] ∆dk+1 = l∈Z qn−3l ∆dkl . n Lemma 3.3. Assume that the subdivision scheme Sq[j+1] is contractive. Then, the scheme Sq[j] is contractive as well. Proof. Consider the case when L = 1 in Eq. (3.8). Assume [j+1] [j+1] [j+1] Sq[j+1] ∞ = max |q3k , |q3k+1 , |q3k+2 , ≤ µ < 1. i∈Z i∈Z i∈Z July 11, S0219691311004249 620 2011 15:46 WSPC/S0219-6913 181-IJWMIP A. Z. Averbuch et al. From Eq. (3.17), [j+1] [j+1] [j+1] [j+1] = ql + ql−1 + ql−2 . a[j+1] (z) = q [j+1] (z) z −2 + z −1 + 1 ⇔ al Hence, [j+1] |a3k |= i∈Z [j+1] |q3k [j+1] i∈Z ≤ i∈Z [j+1] |q3k [j+1] + q3k−1 + q3k−2 | + [j+1] |q3k−1 + i∈Z We have similar estimations for the sums [j+1] |q3k−2 ≤ 3µ. i∈Z [j+1] i∈Z |a3k+1 | [j+1] Sa[j+1] ∞ ≤ 3µ. But, by deﬁnition, q [j] (z) = a µ < 1. The proof for L > 1 is similar. and [j+1] i∈Z |a3k+2 |. Thus, (z)/3 and we have Sq[j] ∞ ≤ Corollary 3.2. Assume that Sa is a TISS, whose symbol a(z) is factorized as −2 m z + z −1 + 1 a[m] (z), a[m] (0) = 3. (3.18) a(z) = 3 If the subdivision scheme Sq[m] is contractive then the schemes Sa[j] , j = 1, . . . , m, and Sa are convergent. Proof. The convergence of Sa follows from Theorem 3.2 and Lemma 3.3. Theorem 3.3. Assume that Sa is a TISS of class P whose symbol is factorized as in Eq. (3.18) and the subdivision scheme Sq[m] is contractive. Then, Sa converges on any initial data f 0 ∈ l1 to the limit function Sa∞ f 0 ∈ C m and dm ∞ 0 S f = Sa∞[m] ∆m f 0 . dxm a Moreover, for j = 1, . . . , m, the scheme Sa[j] with the symbol a[j] (z) = (3/(z −2 + z −1 + 1))j , satisﬁes Sa[j] dj (Sak f 0 ) = dj (Sak+1 f 0 )), Sa∞[j] dj f 0 = dj ∞ 0 S f . dxj a (3.19) Proof. The convergence of Sa and Sa[j] , j = 1, . . . , m, follows from Corollary 3.2. We prove the theorem for m = 1. Assume that the initial data ϕ0 = δ is the Kronecker delta. Then, ϕk = Sak δ and ϕ∞ = ϕ(x), which is the BLF of Sa . From (3.16) dϕk+1 = Sa[1] dϕk . We introduce the sequence of the secondorder splines {g k (x)}k∈Z , x ∈ R, which interpolate the subsequently reﬁned data g k (3−k j) = dϕkj , j ∈ Z, while the initial data is {dϕ0j }. Since Sa[1] is convergent, the sequence of functions {g k (x)}k∈Z at any ﬁnite interval converges uniformly to a limit function g = Sa∞[1] ∆δ ∈ C(R). Deﬁne the piecewise constant function ∆ hk (x) = (dϕk )j , 3−k j ≤ x < 3−k (j + 1), j ∈ Z. July 11, S0219691311004249 2011 15:46 WSPC/S0219-6913 181-IJWMIP Ternary Interpolatory Subdivision Schemes Originated from Splines 621 It is clear that |g k (x) − hk (x)| ≤ ∆dϕk ∞ , to g(x). and, from Eq. (3.13) we get that hk (x) converges uniformly xAll kthe funcx g(t)dt − h (t)dt| ≤ tions considered here decay exponentially and that | −∞ −∞ x x k k |g(t) − h (t)|dt. Thus, −∞ x the sequence { −∞ h (t)dt : k ∈ Z+ } converges uniformly to the function −∞ g(t)dt. From the deﬁnition of hk (x) we get x hk (t)dt = (Sak δ)j ψ(3k x − j), −∞ j∈Z where ψ(x) = x 1 − |x|, x ∈ [−1, 1]. Since Sa is convergent, Lemma (3.2) implies that −∞ hk (t)dt converges uniformly to ϕ(x), and therefore, ϕ(x) = x d ∞ −∞ g(t)dt, dx ϕ(x) = g(x) = Sa[1] ∆δ ∈ C(R). We conclude from (3.14) that for all the initial data f 0 ∈ 1 , (3.19) holds with m = 1. The proof for m > 1 is similar. 4. TISS Derived from Interpolatory Splines In this section, we present two families of TISS of type P, which are derived from the continuous and the discrete interpolatory splines. The subdivision scheme, which originated from continuous splines, is presented in Ref. 14. In the present paper we brieﬂy outline this scheme. The subdivision scheme, which is based on discrete splines, will be described in more details. 4.1. TISS based on continuous splines We denote by Sp the space of polynomial splines σ p (x) of order p (degree p − 1) ∆ deﬁned on the uniform grid g0 = {k} , k ∈ Z, such that the arrays {S p (k)} , k ∈ Z, belong to l1 . The reﬁnement mask of the subdivision scheme is derived from the following: Continuous Spline Triadic Insertion Rule (CSTIR). We construct on the ∆ grid gj = k3−j , j = 0, 1, . . . , k ∈ Z, a spline S j , which interpolates the sequence ∆ f j = {fkj } on the grid gj . Then, fkj+1 = S j (k3−j−1 ), k ∈ Z. In this section, we use a signal processing terminology in which reﬁnement masks designate a ﬁlter. Note that the value of a spline at any point can be expressed as a linear combination of its values at grid points. In other words, any value fkj+1 can be derived by some ﬁltering of the sequence f j . We present explicit expressions for these ﬁlters for splines of arbitrary order. Moreover, it turns out that for any j ∈ N, fkj = S 0 (k3−j ), k ∈ Z. Thus, we obtain a fast algorithm for computing the values of a spline from Sp , which interpolates the sequence f 0 on the grid g0 at triadic rational points {k3−j t}, k ∈ Z. July 11, S0219691311004249 622 2011 15:46 WSPC/S0219-6913 181-IJWMIP A. Z. Averbuch et al. 4.1.1. Interpolatory splines The centered B-spline of the ﬁrst order is the characteristic function of the interval [−1/2, 1/2]. The centered B-spline of order p is the convolution M p (x) = M p−1 (x)∗ M 1 (x), p ≥ 2. The B-spline of order p is supported on the interval (−p/2, p/2). It is positive within its support and symmetric about zero. The B-spline M p consists of pieces of polynomials of degree p − 1 that are linked to each other at the nodes such that M p ∈ C p−2 . Nodes of B-splines of even and odd orders are located at points {k} and {k + 1/2}, k ∈ Z, respectively. The explicit representation of a B-spline is p−1 p−1 p 1 p k p t+ −k (−1) M (t) = k (p − 1)! 2 + k=0 ∆ where t+ = (t + |t|)/2. Shifts of B-splines form a basis in Sp . Namely, any spline σ p (x) ∈ Sp has the following representation: σ p (x) = qk M p (x − k). k∈Z The z-transform of the sampled B-spline ∆ U p (z) = z k M p (k) (4.1) k∈Z is a Laurent polynomial, which is symmetric about inversion. Its roots are real, simple and do not lie on the unit circle |z| = 1 (see Refs. 10 and 11). This polynomial U p (z) can be calculated explicitly for any order p of the splines. Assume that the values of the spline at the grid points S p (k) = fk0 , k ∈ Z. The z-transform is k∈Z z k S p (k) = q(z)U p (z) = f 0 . Thus, the z-transform of the coeﬃcients of the spline, which interpolates the data f 0 is q(z) = f 0 (z) . U p (z) 4.1.2. Subdivision with reﬁnement masks derived from continuous splines When the subdivision is implemented according to CSTIR where continuous splines are involved, the symbol of the schemes can be presented explicitly. p Theorem 4.1.14 The symbol C p (z) of the TISS SC , which is generated by CSTIR with a continuous interpolatory spline of order p, is C p (z) = (z −1 + 1 + z)p U p (z) , 3p−1 U p (z 3 ) (4.2) p where the Laurent polynomial U p (z) is deﬁned in Eq. (4.1). The scheme SC is of type P . July 11, S0219691311004249 2011 15:46 WSPC/S0219-6913 181-IJWMIP Ternary Interpolatory Subdivision Schemes Originated from Splines 623 The symbol C p (z) can be represented in a polyphase form: p C p (z) = z k cpk = zC−1 (z 3 ) + C0p (z 3 ) + z −1 C1p (z 3 ), k∈Z ∆ C0p (z) = ∆ z k cp3k , p C±1 (z) = k∈Z z k cp3k±1 . k∈Z Respectively, the z-domain representation of the subdivision step is k (z 3 ) + f0k (z 3 ) + z −1 f1k (z 3 ), f k+1 (z) = C p (z)f k (z 3 ) = zf−1 k p fm (z) = Cm (z)f k (z), m = −1, 0, 1. p is interpolatory, therefore, C0p (z) ≡ 1. Due to the symmetry of The scheme SC p p p (z −1 ). U (z), C1 (z) = C−1 4.1.3. Examples of symbols Linear spline: 2 (z −1 ) = C12 (z) = C−1 2 C (z) = z −1 C12 (z 3 ) z+2 , 3 +1+ 2 zC−1 (z 3 ) 2 z + 1 + z −1 . = 3 (4.3) This is a single symbol in the family, whose mask is ﬁnite. All the masks, which were derived from splines of higher orders, are inﬁnite but exponentially decaying. Quadratic spline: 3 z + 6 + z −1 z + 1 + z −1 C (z) = , 9 (z 3 + 6 + z −3 ) 3 C13 (z) = 3 C−1 (z −1 ) 25z + 46 + z −1 . = 9 (z + 6 + z −1 ) Cubic spline: 4 z + 4 + z −1 z + 1 + z −1 C (z) = , 27 (z 3 + 4 + z −3 ) 4 4 C14 (z) = C−1 (z −1 ) = z 2 + 60z + 93 + 8z −1 . 27 (z + 4 + z −1 ) Spline of fourth degree: 5 2 z + 76z + 230 + 76z −1 + z −2 z + 1 + z −1 5 C (z) = , 81 (z 6 + 76z 3 + 230 + 76z −3 + z −6 ) 5 C15 (z) = C−1 (z −1 ) = 625z 2 + 11516z + 16566 + 2396z −1 + z −2 . 81 (z 2 + 76z + 230 + 76z −1 + z −2 ) (4.4) July 11, S0219691311004249 624 2011 15:46 WSPC/S0219-6913 181-IJWMIP A. Z. Averbuch et al. Spline of ﬁfth degree: 6 2 z + 26z + 66 + 26z −1 + z −2 z + 1 + z −1 6 , C (z) = 243 (z 6 + 26z 3 + 66 + 26z −3 + z −6 ) 6 (z −1 ) = C16 (z) = C−1 z 3 + 1018z 2 + 10678z + 14498 + 29336z −1 + 32z −2 . 243 (z 2 + 26z + 66 + 26z −1 + z −2 ) p Convergence: An important fact about TISS SC originated from continuous interpolatory splines is established in Ref. 14. Namely, it was proved that values of the spline σ p (x) at any set of triadic rational points σ p (k3−j ) can be calculated by successive applications of the CSTIR to the initial data array f 0 . Theorem 4.2.14 Assume that the spline σ0p (x) belongs to Sp , σ0p (k) = fk0 and f 0 = {fk0 } ∈ l1 . Let C p be the mask, whose symbol is deﬁned by Eq. (4.2). Assume that for j ∈ N, f j = {fkj } is the array whose z-transform is derived from the relation F j+1 (z) = C p (z)F j (z 3 ). Then, σ0p (k3−j ) = fkj , j ∈ N. p Corollary 4.1. The TISS SC , which is generated by CSTIR with the continuous interpolatory spline of order p, converges to the spline σ0p (x), which interpolates the initial data array σ0p (k) = fk0 . Recall that splines of order p have p − 2 continuous derivatives. Remark. If the initial data is the delta sequence fk0 = δ(k) Lp (k3−l ) where Lp (x) is the fundamental spline in the space p . It decays exponentially. other hand, is BLF of the TISS SC extend the assertion of Theorem 4.2 from splines belonging to interpolate sequences of power growth. then we get fkj = Sp , which, on the Therefore, we can Sp to splines that Remark. Theorem 4.2 provides an eﬃcient algorithm for fast calculation of the interpolatory spline at triadic rational points. 4.2. Refinement masks originated from discrete splines In this section, we use a special type of the so-called discrete splines for the design of the reﬁnement mask. The discrete splines are deﬁned on the grid {k}k∈Z and they are the counterparts of continuous polynomial splines. The discrete splines were used in Refs. 7–6 for the construction of wavelets and wavelet frames. The reﬁnement mask of the subdivision scheme is derived from the following: Discrete Spline Triadic Insertion Rule (DSTIR). On the grid {k}k∈Z , we construct the discrete spline dpj (k) of order p with the base N = 3 such that dpj (3k) = fkj , k ∈ Z. Then, j+1 = dpj (3k + r), f3k+r r = −1, 0, 1, k ∈ Z. (4.5) July 11, S0219691311004249 2011 15:46 WSPC/S0219-6913 181-IJWMIP Ternary Interpolatory Subdivision Schemes Originated from Splines 625 4.2.1. Discrete B-splines The sequence 1 BN (j) = 1 0 if j = 0, . . . , N − 1, N ∈ N, otherwise is called the discrete B-spline of the ﬁrst order, whose base is N . Higher-order p p−1 1 B-splines are deﬁned as discrete convolutions by the recurrence BN = BN ∗ BN . Obviously, the z-transform of the B-spline of order p is p (z) = (1 + z −1 + z −2 + · · · + z −(N −1) )p , BN p = 1, 2, . . . . Since N = 2M + 1 is odd, then we can introduce the so-called the centered B-spline to be ∆ p p p p (l)}, rN (l) = BN (l + M p) ⇔ RN (z) = (z M + · · · + z −M )p . rpN = {rN 4.2.2. Interpolatory discrete splines In our construction, we use discrete splines with the base N = 3 and drop the index ·N in the notation of the B-spline. Thus, rp (l) and Rp (z) will stand for p p (l) and RN (z), respectively. Similarly to continuous splines, a discrete spline rN p p d = {d (l)}l∈Z of order p on the grid {3n}n∈N is deﬁned as a linear combination with real-valued coeﬃcients of shifts of the centered B-splines: ∆ dp (l) = ∞ hn rp (l − 3n) ⇔ Dp (z) = H(z 3 )Rp (z). n=−∞ ∆ Denote by rrp (l) = rp (r +3l), r = −1, 0, 1, the polyphase components of the discrete ∆ B-spline. Then, the polyphase components dpr (l) = dp (r + 3l), r = −1, 0, 1, of the discrete spline dp (l) are ∆ dpr (l) = ∞ hn rrp (l − 3n) ⇔ Drp (z) = H(z)Rrp (z), r = −1, 0, 1. (4.6) n=−∞ Proposition 4.1.7 The component R0p (z) is symmetric about the inversion z → 1/z and positive on the unit circle |z| = 1. All the components Rrp (z) have the same value at z = 1: Rrp (1) = R0p (1), r = −1, 1. (4.7) The scheme for designing the reﬁnement masks, which uses discrete splines, is similar to the scheme that is based on continuous splines. We construct the discrete spline dp , which interpolates the data x = {x(l)}l∈Z on the sparse grid {3l} and calculate the values of the constructed spline at the points {3l ± 1}. Using Eq. (4.6), July 11, S0219691311004249 626 2011 15:46 WSPC/S0219-6913 181-IJWMIP A. Z. Averbuch et al. we ﬁnd the z-transform of the coeﬃcients of an interpolatory spline to be dp0 (l) = x(l) ⇔ D0p (z) = H(z)R0p (z) = X(z) ⇒ H(z) = X(z) . R0p (z) (4.8) The z-transforms of dpr is Drp (z) = H(z)Rrp (z) = Trp (z)X(z), ∆ Trp (z) = Rrp (z) . R0p (z) (4.9) Equation (4.7) implies that Trp (1) = 1. In order to calculate the polyphase components Rrp (z), we have to solve the following system: p (z 3 ) + z −1 · R1p (z 3 ), Rp (z) = R0p (z 3 ) + z · R−1 p Rp (z · e2πi/3 ) = R0p (z 3 ) + ze2πi/3 · R−1 (z 3 ) + z −1 e−2πi/3 · R1p (z 3 ), p (z 3 ) + z −1 e2πi/3 · R1p (z 3 ). Rp (z · e−2πi/3 ) = R0p (z 3 ) + ze−2πi/3 · R−1 Thus, R0p (z 3 ) = Rp (z) + Rp (z · e2πi/3 ) + Rp (z · e−2πi/3 ) , 3 p R−1 (z 3 ) = z −1 Rp (z) + Rp (z · e2πi/3 )e−2πi/3 + Rp (z · e−2πi/3 )e2πi/3 , 3 (4.10) Rp (z) + Rp (z · e2πi/3 )e2πi/3 + Rp (z · e−2πi/3 )e−2πi/3 . 3 We deﬁne the symbol of the reﬁnement mask of TISS derived from the discrete splines of degree p to be R1p (z 3 ) = z p T p (z) = zT−1 (z 3 ) + 1 + z −1 T1p (z 3 ) where p T−1 (z 3 ) p R−1 (z 3 ) Rp (z) + Rp (z · e2πi/3 )e−2πi/3 + Rp (z · e−2πi/3 )e2πi/3 = p 3 = z −1 , R0 (z ) Rp (z) + Rp (z · e2πi/3 ) + Rp (z · e−2πi/3 ) T1p (z 3 ) = Rp (z) + Rp (z · e2πi/3 )e2πi/3 + Rp (z · e−2πi/3 )e−2πi/3 R1p (z 3 ) p 3 =z R0 (z ) Rp (z) + Rp (z · e2πi/3 ) + Rp (z · e−2πi/3 ) p = T−1 (z −3 ), Rp (z) = (z + 1 + 1/z)p . Hence, it follows that 2π 3Rp (z) + Rp (z · e2πi/3 ) + Rp (z · e−2πi/3 ) 1 + 2 cos 3 T p (z) = Rp (z) + Rp (z · e2πi/3 ) + Rp (z · e−2πi/3 ) = 3(z + 1 + 1/z)p p p . (z + 1 + 1/z)p + e2πi/3 z + 1 + e−2πi/3 /z + e−2πi/3 z + 1 + e2πi/3 /z Proposition 4.1 implies that the derived subdivision schemes belong to Class P. July 11, S0219691311004249 2011 15:46 WSPC/S0219-6913 181-IJWMIP Ternary Interpolatory Subdivision Schemes Originated from Splines 627 4.2.3. Examples of reﬁnement masks derived from discrete splines Linear TISS, p = 2: 2 T1,3 (z) = T 2 (z) = 2+z , 3 2 T−1,3 (z) = 2 + 1/z , 3 (z + 1 + z −1 )2 . 3 Quadratic TISS, p = 3: 3 T1,3 (z) = T 3 (z) = 3(2 + z) , z + 7 + 1/z 3 T−1,3 (z) = 3(2 + 1/z) , z + 7 + 1/z (z + 1 + z −1 )3 . z −3 + 7 + z 3 Cubic TISS, p = 4: 4 (z) = T1,3 T 4 (z) = 1/z + 16 + 10z , 4z + 19 + 4/z 4 T−1,3 (z) = z + 16 + 10/z , 4z + 19 + 4/z (z + 1 + z −1 )4 . 4z −3 + 19 + 4z 3 5. Implementation of TISS Originated from Splines From the signal processing point of view, implementation of TISS is an iterated ﬁltering coupled with the upsampling of the initial data sequence f 0 . The derived symbols of the subdivision masks serve as the transfer functions of the ﬁlters. The masks serve as the impulse responses (IR) of the ﬁlters. All the derived ﬁlters except for the ﬁlters derived from the second-order splines have inﬁnite IR (IIR). But, since the transfer functions are rational and do not have poles on the unit circle, ﬁltering can be implemented in a fast recursive way. We brieﬂy outline the implementation procedures. In Ref. 3, it is discussed in more details. There are two ways to implement the derived subdivision schemes: polyphase and direct ﬁltering. 5.1. Polyphase filtering One way is to implement the ﬁlter Φ = {Φk }, k ∈ Z, is by using the so-called polyphase representation of the ﬁlter: ∆ z −k Φ3k+r , r = −1, 0, 1. Φ(z) = zΦ−1 (z 3 ) + Φ0 (z 3 ) + z −1 Φ1 (z 3 ), Φr (z) = k∈Z Then, the polyphase representation of the array f j+1 is j+1 3 F j+1 (z) = zF−1 (z ) + F0j+1 (z 3 ) + z −1 F1j+1 (z 3 ), ∆ j+1 z −k f3k+r = Φr (z)F j (z), r = −1, 0, 1. Frj+1 (z) = k∈Z July 11, S0219691311004249 628 2011 15:46 WSPC/S0219-6913 181-IJWMIP A. Z. Averbuch et al. j+1 Thus, in order to retrieve the sub-arrays {f3k±1 }, we have to ﬁlter the array j {fk } with the ﬁlters whose transfer functions are Φ±1 (z), respectively. Recall that j+1 {f3k = fkj }. Example. Implementation of the Filter C3 . The z-transform of these ﬁlters given in Eq. (4.4) is: C 3 (z) = α 25z + 46 + z −1 25z + 46 + z −1 = , −1 9 (z + 6 + z ) 9 (1 + αz)(1 + αz −1 ) √ where α = 3 − 2 2 ≈ 0.172. Then, application of the ﬁlter T31 to a data array f = {fk }, whose z-transform is F (z), is implemented as a subsequent application of the three ﬁlters: C3 f = Ψl Ψr Ψ · f. The ﬁlters are deﬁned by their z-transforms: Ψ(z) = α 25z + 46 + z −1 , 9 Ψr (z) = 1 , 1 + αz −1 Ψl (z) = 1 . 1 + αz Thus, ﬁltering is carried out in three steps: α F 1 (z) = Ψ(z)F (z) ⇔ fk1 = (25fk+1 + 46fk + fk−1 ) , 9 2 1 2 F (z) = Ψr (z)F (z) ⇔ 1 + αz −1 F 2 (z) = F 1 (z) ⇔ fk2 = fk1 − αfk−1 , G(z) = Ψl (z)F 2 (z) ⇔ (1 + αz) G(z) = F 2 (z) ⇔ gk = fk2 − αgk+1 . The ﬁlter Ψ has ﬁnite IR (FIR) unlike the ﬁlters Ψl and Ψr . Application of the ﬁlter Ψr is called a causal recursive ﬁltering. Here, for the calculation of the term fk2 , the 2 is exploited. Application of Ψl is called anti-causal previously derived term fk−1 recursive ﬁltering. All these procedures are implemented in a fast way. Computation of splines of higher orders uses ﬁlters, which are factorized into longer cascades of the same structure. How to choose the initial values for recursive ﬁltering was shown in Ref. 7. Here we use this result. f11 ≈ f1 + d (−α)n fn n=1 1 gN ≈ fN + d (−α)n fN −n n=1 where d is the prescribed initialization depth. July 11, S0219691311004249 2011 15:46 WSPC/S0219-6913 181-IJWMIP Ternary Interpolatory Subdivision Schemes Originated from Splines 629 5.2. Direct filtering Equation (4.2) suggests that, when several steps of the subdivision with continuous splines are carried out, a direct application of the ﬁlter Tp is preferable. It follows from Eq. (4.2) that F j+1 (z) = C p (z) · F j (z 3 ) = j−1 l j C p (z 3 ) · F 0 (z 3 ) l=1 U p (z) = j−1 l l (z −3 + 1 + z 3 )p j l=0 U p (z 3j ) · F 0 (z 3 ). For example, F 3 (z) = U p (z)(z −1 + 1 + z)p (z −3 + 1 + z 3 )p (z −9 + 1 + z 9 )p · F 0 (z 27 ). U p (z 27 ) Thus, the subdivision is implemented via the following steps: (1) The IIR ﬁlter with the transfer function 1/U p (z) is applied to the data array f 0 . (2) The produced array is upsampleda and ﬁltered with a FIR ﬁlter, whose transfer function is (z −1 + 1 + z)p (repeated j times). (3) The produced array is ﬁltered with a FIR ﬁlter whose transfer function is U p (z). Note that in this case, the IIR ﬁltering is applied only once. 6. Examples of Spline-Based TISS In this section, we provide the results about the convergence and smoothness of the TISS that were originated from quadratic and cubic discrete splines. Quadratic interpolatory discrete spline. The symbol of the TISS Sa is a(z) = (z + 1 + z −1 )3 . z 3 + 7 + z −3 The symbol of the appropriate scheme of diﬀerences is: q(z) = z −2 a(z) . + z −1 + 1 The algorithm, which veriﬁes the convergence of TISS, is used. We get Sq1 = 0.6 < 1, therefore, the quadratic TISS Sa is convergent. From the algorithm, which veriﬁes the rate of smoothness, we get Sq1[1] = 0.6 < 1. Thus, Sa ∈ C 1 . a Upsampling means replacing an array {ak } by the array {ãk } such that ã3k = ak and ã3k±1 = 0. July 11, S0219691311004249 630 2011 15:46 WSPC/S0219-6913 181-IJWMIP A. Z. Averbuch et al. Fig. 1. The basic limit functions: Left: Quadratic discrete spline TISS; Center: Cubic discrete spline TISS; Right: Linear spline TISS. Cubic interpolatory discrete spline. The symbol of the TISS Sa is a(z) = (z + 1 + z −1 )4 . 4z 3 + 19 + 4z −3 The symbol of the appropriate scheme of diﬀerences is: q(z) = a(z) . z −2 + z −1 + 1 Since Sq1 = 0.52223 < 1, the cubic TISS Sa is convergent. From the algorithm, which veriﬁes the rate of smoothness, we get Sq1[2] = 0.81818 < 1. Thus, Sa ∈ C 2 . Appendix. Evaluation of Coeﬃcients of Subdivision Masks via the Discrete Fourier Transform Assume L = 1 in Eq. (3.9). L > 1 is treated similarly. Assume that N = 2p , p ∈ N. N/2 The Discrete Fourier Transform (DFT) of an array xp = {xpk }k=−N/2 and its inverse (IDFT) are N/2 x̂pn = e−2πikn/N xpk and xpk = k=−N/2 1 N N/2 e2πikn/N x̂pk . (A.1) k=−N/2 We assume that z = e−iw . The coeﬃcients of the masks decay exponentially, i.e. ∞ ∞ ∞ N −1 |ak | ≤ Aγ k ⇒ |ak | ≤ Aγ k = A γk − γk k=N =A k=N N −1 γ γ(1 − γ − 1−γ 1−γ k=1 ) k=1 = Bγ N (A.2) July 11, S0219691311004249 2011 15:46 WSPC/S0219-6913 181-IJWMIP Ternary Interpolatory Subdivision Schemes Originated from Splines 631 where B = A/(1 − γ), 0 < γ < 1 and A is some positive constant. We need to evaluate the sums ∞ Sa,p = |a3k+p |, p ∈ {−1, 0, 1}. k=−∞ Denote A(θ) = a(e−iθ ) = ∞ T (e−iθ ) = ak e−iθk . P (e−iθ ) k=−∞ The function A is calculated at the discrete set of points ∞ ∞ 2πn e−2πikn/N ak = e−2πin(k+lN )/N ak ân = A = N k=−∞ N/2−1 = e−2πirn/N ar+lN l=−∞ r=−N/2 ∞ N/2−1 = e ∞ ∞ −2πirn/N ar+lN l=−∞ r=−N/2 ϕr = k=−∞ l∈Z ar+lN = ar + ψr , ψr = l=−∞ ar+lN l∈Z\0 From Eq. (A.2) we get |ψr | ≤ 2BαN ⇒ |ar | = |ϕr | + αN r , N |αN r | ≤ 2Bγ . (A.3) The samples ϕk are derived from the application of IDFT: ϕk = 1 N N/2−1 e2πikn/N ân . n=−N/2 By using Eq. (A.3), we can evaluate the sums that we are interested in as follows: N/4−1 Sa,p = |a3k+p | + 2 k=−N/4 k=−N/4 |αN 3k+p | + 2 ∞ N/4−1 |a3k+p | = k=N/4 N/4−1 ρN = ∞ |ϕ3k+p | + ρN , k=−N/4 |a3k+p |, |ρN | ≤ B(N + 2)γ N . k=N/4 Hence, it follows that by doubling N we can approximate the inﬁnite series Sa,p by N/4−1 N = k=−N/4 |ϕ3k+p |, whose terms are derived from the applithe ﬁnite sum σa,p cation of the DFT. An appropriate value of N can be derived theoretically using July 11, S0219691311004249 632 2011 15:46 WSPC/S0219-6913 181-IJWMIP A. Z. Averbuch et al. estimations on the roots in the denominator. Practically, we can iterate the cal2N becomes culations by gradually doubling N until the result from calculating σa,p N identical to σa,p (up to a machine precision). The same approach is valid for the (L) evaluation of the sums j |q3L ·j−i | for any L. The algorithm that veriﬁes the convergence of Sa with the mask a(z): (1) Compute: q(z) = a(z)/(z −2 + z −1 + 1). (1) (2) Set qn = q(e−2πin/N ), N = 2p , p ∈ N, −N/2 ≤ n ≤ N/2 − 1. (3) Verify that SqL is contractive: for L = 1, . . . , M (a) for i = 0, . . . , 3L − 1 (L) N/4−1 3L −1 Compute {Ki }i=0 : Ki = j |q3L ·j−i | r=−N/4 |ϕ3L r−i |, N/2−1 [L] where ϕk = N −1 n=−N/2 e2πnki/N qn for N sufficiently large. (b) If max{Ki } < 1, 0 ≤ i ≤ 3L − 1 then Sq is contractive, therefore Sa is convergent. Stop! else (L+1) = q (L+1) (e−2πin/N ), compute qn L+1 j−1 (L+1) (z) = j=1 q(z 3 ), N = 2p , p ∈ N, −N/2 ≤ n ≤ N/2 − 1. where q (4) If after M iterations Sq is not contractive. Stop! References 1. N. Dyn, J. A. Gregory and D. Levin, Analysis of uniform binary subdivision schemes for curve design, Constr. Approx. 7 (1991) 127–147. 2. N. Dyn, Analysis of convergence and smoothness by the formalism of Laurent polynomials, in Tutorials on Multiresolution in Geometric Modelling, eds. A. Iske, E. Quak and M. S. Floater (Springer, 2002), pp. 51–68. 3. V. A. 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