Computation of interpolatory splines via triadic subdivision

Transcription

1 Computation of interpolatory splines via triadic subdivision Valery A. Zheludev and Amir Z. Averbuch Abstract We present an algorithm for computation of interpolatory splines of arbitrary order at triadic rational points. The algorithm is based on triadic subdivision of splines. Explicit expressions for the subdivision symbols are established. These are rational functions. The computations are implemented by recursive filtering. 1 Introduction Denote by S p the space of polynomial splines S p x of order p defined on the uniform grid g = {k}, k Z, such that the arrays {S p k}, k Z, belong to l 1. We propose to compute values of the splines via a triadic subdivision. The insertion rule for a spline Sx S p is Triadic Insertion Rule: Let f { = f k = S p k }, k Z. For j =, 1,..., we construct on the grid g j = { k3 j} }, k Z, a spline S p j x, which interpolates the sequence fj = {f j k on the grid g j. Then, = S p j k3 j+1, k Z. f j+1 k 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 f j+1 k can be derived by some filtering of the sequence f j. We present explicit expressions for these filters for splines of arbitrary order. Their transfer functions are rational functions. Computations are implemented by recursive filtering. Moreover, we prove that for any j N, f j k = S k3 j, k Z. Thus, we obtain a fast algorithm that computes the values of a spline S p x from the space Sp, which interpolates the sequence f on the grid g, at the triadic rational points { k3 j}, k Z. 2 Spline filters 2.1 B-splines The centered B-spline of first order is the characteristic function of the interval [ 1/2, 1/2]. The centered B-spline of order p can be expressed as the convolution M p x = M p 1 x M 1 x, p 2. Note that the B-spline of order p is supported on the interval p/2, p/2. It is positive within its 1

2 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 order are located at points {k} and of odd order at points {k + 1/2}, k Z. The Fourier transform of the B-spline of order p is M p ω = sin ω/2 p e iωx M p x dx =. 2.1 ω/2 The time domain representation of the B-spline is M p x = 1 p 1 p 1 k p 1! k k= x + p p 1 2 k, x + = x + x /2. + Shifts of B-splines form a basis in the space S p. Namely, any spline S p x S p has the following representation: Equation 2.1 implies that for any p 2 M p x k = 1 2π = e 2πiωk m p xω dω, S p x = e iωx k sin ω/2 ω/2 q k M p x k. p dω = e 2πilx l Z where m p xω = e 2πiωx sin πω p l Z e 2πiωx k sin πωp πl + ω p e 2πilx dω πl + ω p. 2.2 The relation 2.2 means that M p x k is a Fourier coefficient of the 1-periodic function m p xω and this function can be represented as the sum: m p xω = e2πikω M p x k. The functions m p xω are 1-periodic with respect to the variable ω and are splines from S p with respect to x. Their shifts along the grid are m p x+l ω = e 2πikω M p x k l = e 2πilω e 2πikω M p x k = e 2πilω m p xω. Following Schoenberg [4], we call the splines m p xω the exponential splines. 2.2 Interpolatory splines Equation 2.2 implies that a spline S p x S p can be represented via the exponential splines: S p x = q k M p x k = q k e 2πiωk m p xω dω = ˆqωm p xω dω, 2.3 where ˆqω = e 2πiωk q k is the discrete Fourier transform DFT of the coefficient sequence q = {q k }, k Z. Integral representation 2.3 was established in [5]. Denote u p ω = m p ω = e 2πikω M p k

3 Due to the compact support and symmetry of the B-splines, the functions u p ω are cosine polynomials. They were extensively studied in [1]. In particular, [1] established that u p ω > for all ω R and p N. In addition, recurrent relations for computation of the polynomials of any order were presented in [1]. However, u p ω can be easily calculated directly from Eq Equation 2.2 provides the following representation of the functions u p ω : u p ω = sin πω p l Z πl + ω p. The values of the spline S p x at the grid points and their DFT are S p n = ˆqωm p nω dω = e 2πiωnˆqωu p ω dω Ŝp ω = ˆqωu p ω. Assume that the spline S p x interpolates a sequence f at the grid points S p n = f n f ω = Ŝp ω ˆqω = f ω u p ω. 2.5 According to Triadic Insertion Rule, which was formulated in Section 1, f 1 n = S p n/3. Then, f 1 ω = n Z e 2πiωn S p n/3 = e 2πiω σ 1 3ω + σ 3ω + e 2πiω σ 1 3ω, 2.6 σ ω = n Z e 2πiωn S p n = ˆqωu p ω = f ω, σ ±1 ω = n Z e 2πiωn S p n ± 1. 3 Due to Eq. 2.2, S p n ± 1 = 3 ˆqωm p n±1/3 ω dω = v p ±1 ω = m p ±1/3 ω = e±2πiω/3 sin πω p l Z e 2πiωnˆqωv p 1 ω dω, e ±2πil/3 πl + ω p. 2.7 Hence, using Eq. 2.5, we obtain σ ±1 ω = vp ±1 ω f u p ω ω Triadic subdivision 3.1 Structure of filters Substituting Eq. 2.8 into Eq. 2.6, we obtain f 1 ω = T p ω f 3ω, T p ω = e 2πiω vp 1 3ω u p 3ω e 2πiω vp 1 3ω u p 3ω. 3.9 Similarly, f j+1 ω = T p ω f j 3ω, j N. Thus, in order to produce the array f j+1, we need to filter the upsampled array f j with the filter T p, whose DFT the frequency response T p ω is given in Eq We establish now the structure of the filter T p. 3

4 Theorem 3.1 The DFT of the filter T p, which originates from the spline of order p, is T p ω = cos πωp u p ω 3 p 1 u p ω Proof: We have from Eqs. 2.7 and 3.9 T p ω = G p ω/u p 3ω, where G p ω = e 2πiω v p 1 3ω + up 3ω + e 2πiω v p 1 3ω = sin 3πωp l Z = cos πω p sin πω p l Z cos 2πl/3. πl + 3ω p 1 + e 2πil/3 + e 2πil/3 πl + 3ω p We split the sum in the last equation into three sub-sums along l = 3m, l = 3m + 1 and l = 3m 1. Respectively, the function G p ω splits as G p ω = cos πω p γ ω + γ 1 ω + γ 1 ω, where γ ω = γ ±1 ω = Hence, Eq. 3.1 follows. sin πωp 3 p m Z sin πωp 3 p m Z 1 mp πm + ω p cos 2πm = 3 p+1 u p ω, 1 m+1p cos 2πm ± 1/3 =. πm + ω ± π/3 p 3.2 Convergence The spline L p x S p, which interpolates the Kronecker delta sequence δk, is called the fundamental spline of the space S p. The spline S p x S p, which interpolates the data sequence f = { f k }, can be represented as S p x = f k Lp x k L p x decays exponentially when x grows. Therefore, Eq represents the spline, which interpolates the sequence { f k }, k Z, even in the case when the sequence has a power growth. In this case, the spline S p x also has a power growth. It was proved in [2] that the spline of power growth, interpolating the data of power growth, is unique. We prove that the values of the spline S p x at any set of triadic rational points S p k3 j can be calculated by successive application of Triadic Insertion Rule, which was formulated in Section 1, to the array f. A similar fact holds for Dyadic Insertion Rule when order p of splines is even. But this is not true for Dyadic Insertion Rule when p is odd. In this case, when p is odd, the spline-based subdivision no longer converges to a spline but rather to a function, which is smoother than the generating spline see [6]. 4

5 Theorem 3.2 Assume the spline S p x belongs to Sp, S p k = f k and f = { fk } l1. Let T p be { } the filter defined by Eq Assume that for j = 1, 2,..., f j = f j k is the array, whose DFT is found from the relation f j+1 ω = T p ω f j 3ω. Then, S p k3 j = f j k, j = 1, 2,..., Proof: We have S p k3 1 = f 1 k from the definition of Triadic Insertion Rule. Next step of subdivision consists in the construction of the spline S p 1 x on the grid g1 = { k3 1}, such that S p 1 k3 1 = f 1 k. Then, fk 2 = Sp 1 k3 2. We prove that fk 2 = Sp k/9. The subsequent relations in Eq are derived by a simple induction. The array f 2 is obtained by repeated application of the filter T p to the array f : f 2 ω = T p ω f 1 3ω = T p ω T p 3ω f 9ω = cos πωp u p ω cos 3πω p u p 3ω 3 p 1 u p 3ω 3 p 1 u p 9ω f 9ω = Hω f u p 9ω 9ω, where Hω = 9 1 p cos πω1 + 2 cos 3πω p u p ω. Denote s k = S p k/9, k Z. Then, the DFT of this array is ŝω = H f 9ωω/u p 9ω, where Hω = 4 e 2νπiω w ν 9ω + e 2νπiω w ν 9ω + u p 9ω, w ±ν ω = m p ±ν/9 ω. ν=1 We can write, using Eq. 2.7, where Hω = sin 9πω p l Z πl + 9ω p e 2νπil/9 + e 2νπil/9 ν=1 = cos 3πω1 + 2 cos πω p sin πω p l Z = cos 3πω1 + 2 cos πω p 4 κ µ ω = µ= 4 4 cos 2πνµ/9 ν=1 κ µ ω, sin πω p 9 p πl + 9ω p m Z 4 cos 2πνl/9 ν=1 1 m+1p πm + ω + µπ/9 p. It is easily verified that for µ = ±1,..., ±4 κ µ ω, while κ ω = 9 1 p u p ω. Hence, it follows that Hω = Hω f 2 k = Sp k3 2, k Z. Repeating our reasoning with the initial data set f 1 instead of f, we prove that f 3 k = Sp 1 k3 3 and so on. 5

6 Remark If the initial data is the delta sequence fk = δ k then we get f j k = Lp k3 l, where L p x is the fundamental spline of the space S p. Therefore, we can extend the assertion of Theorem 3.2 from the splines belonging to S p to the splines that interpolate sequences of power growth. 4 Remarks on the computational aspects of the spline subdivision It was stated in Section 3.1 that in order to produce the array f j+1 we need to filter the upsampled array f j with the filter T p. In the frequency domain, filtering is presented by Eq The frequency response of the filter is given by Eq For computational purposes, we present filtering using the z-transform. We recall that the the z-transform of a sequence a = {a k } l 1 is Az = z k a k. The z-transform of the shifted sequence is z k a k+l = z l Az. Denote z = e 2πiω. Then, the DFT of a sequence a becomes its z-transformâω = Az. The DFT of a filter T becomes its z-transform T ω = T z and it is called the transfer function. The subdivision is represented as F j+1 z = T p zf j z 3, T p z = z z p U p z U p z 3, U p z = z k M p k The function U p z is a Laurent polynomial and the transfer function T p z is a rational function. Such filters are called the infinite impulse response IIR filters as opposite to the finite impulse response FIR filters, whose transfer functions are Laurent polynomials. Proposition 4.1 [2] The roots of the Laurent polynomials U p z are all simple and negative. Each root ζ can be paired with a dual root θ such that ζ θ = 1. Thus, if p = 2r 1, p = 2r then U p z can be represented by: U p z = r n=1 1 γ n 1 + γ n z1 + γ n z 1, < γ 1 < γ 2 <... γ r = e g < 1, g >. The above properties of the polynomials U p z enable to implement filtering via a recursive algorithm, which is widely used in signal processing. There are two ways to implement the described subdivision scheme. Polyphase filtering: One way is to implement the filter T p = {T k }, k Z, using the so-called polyphase representation of the filter: T p z = zt p 1 z3 + T p z3 + z 1 T p 1 z3, Tν p z = z k T p 3k+ν, ν = 1,, 1. Then, the polyphase representation of the array f j+1 is F j+1 z = zf j+1 1 z3 + F j+1 z 3 + z 1 F j+1 1 z 3, Fν j+1 z = z k f j+1 3k+ν = T ν p zf j z, ν = 1,, 1. 6

7 { } { Thus, in order to retrieve the sub-arrays f j+1 3k±1 we have to filter the array { whose transfer functions are T p ±1 z, respectively. Recall that f j+1 3k = f j k } f j k with the filters, }. The polyphase components of the transfer function T p z can be derived from the expression for T p z given in Eq Obviously, T p 1 z = T p 1 z 1. However, it is easily seen from Eq. 3.9 that T p ±1 e2πiω = v p ±1 ω/up ω. Direct filtering: Equation 4.13 suggests that, when several steps of the subdivision must be carried out, a direct application of the filter T p is preferable. It follows from Eq that j 1 F j+1 z = T p z F j z 3 = For example, l=1 T p z 3l F z 3j = U p z j 1 l= z 3l z 3l p U p F z 3j. z 3j F 3 z = U p zz z p z z 3 p z z 9 p U p z 27 F 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. 2. The produced array is upsampled 1 and filtered with FIR filter, whose transfer function is z z p repeated j times. 3. The produced array is filtered with FIR filter whose transfer function is U p z. Note that the IIR filtering is applied only once. Examples of filters Linear spline: T 2 z = z z 1 2 /3, T 2 1 z = T 1 2 z 1 = z + 2/3. This is a single FIR filter in the family. All the filters derived from splines of higher orders are IIR. Quadratic spline: T 3 z = z z 1 z z z z 3, T1 3 z = T 3 1z 1 = 25z z 1 9 z z Cubic spline: T 4 z = z z 1 z z z z 3, T1 4 z = T 1z 4 1 = z2 + 6z z 1 27 z z 1. 1 Upsampling means replacing an array {a k } by the array {ã k } such that ã 3k = a k and ã 3k±1 =. 7

8 Spline of fourth degree : z T z z 1 + z 2 z z 1 5 z = 81 z z z 3 + z 6, T1 5 z = T 1z 5 1 = 625z z z 1 + z 2 81 z z z 1 + z 2. Spline of fifth degree : z T z z 1 + z 2 z z 1 6 z = 243 z z z 3 + z 6, T1 6 z = T 1z 6 1 = z z z z z z z z 1 + z 2. We now provide an example of a recursive implementation of an IIR filter. In [6] this is discussed in more details. Implementation of the filter T 3 1. The z-transform of this filter is given in Eq. 4.14: T 3 1 z = 25z z 1 9 z z 1 = α 25z z αz1 + αz 1, where α = Then, application of the filter T 3 1 to an array a = {a k }, whose z-transform is Az, is implemented as a subsequent application of three filters: b = T 3 1a = R l R r F a, which are are defined by their z-transforms: F z = α 9 25z z 1, R r z = Thus, filtering is carried out in three steps: αz 1, R lz = αz. A 1 z = F zaz a 1 k = α 9 25a k a k + a k 1, A 2 z = R r za 1 z 1 + αz 1 A 2 z = A 1 z a 2 k = a1 k αa2 k 1, Bz = R l za 2 z 1 + αz Bz = A 2 z b k = a 2 k αb k+1. The filter F is FIR unlike the filters R l and R r. Application of the filter R r is called causal recursive filtering. Here, for the calculation of the term a 2 k, the previously derived term a2 k 1 is used. Application of the R l is called anti-causal 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. References [1] I. J. Schoenberg, Contribution to the problem of approximation of equidistant data by analytic functions, Quart.Appl. Math., , 45-99,

9 [2] I. J. Schoenberg, Cardinal interpolation and spline functions, J. Approx. Th., 2, 1969, [3] I. J. Schoenberg, Cardinal interpolation and spline functions II, J. Approx. Theory, , [4] I. J. Schoenberg, Cardinal spline interpolation, CBMS, 12, SIAM, Philadelphia, [5] V. A. Zheludev, Integral representation of slowly growing equidistant splines, Approximation Theory and Applications, 14, no. 4, 1998, [6] V. A. Zheludev, Interpolatory subdivision schemes with infinite masks originated from splines, Advances in Comp. Math. 25,26,