A ternary interpolatory subdivision schemes originated from splines

A ternary interpolatory subdivision schemes originated from splines
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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
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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
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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.
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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)
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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.
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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.
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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)
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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
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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
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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.
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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.
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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 .
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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)
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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)
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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),
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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.
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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
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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.
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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.
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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)
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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
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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!
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