Interpolating Spline Wavelet Packets

Interpolating Spline Wavelet Packets Jianzhong Wang November 16, 2000 Abstract Interpolating wavelets are used to construct the bases for smoothness spaces and for which the coefficients are obtained by sampling rather than integration. We construct the wavelet packets corresponding to the interpolating spline wavelets and provide the interpolating schemes for performing the wavelet transforms.

1

Introduction

Let Cu be the Banach space of the functions, which are bounded and uniformly continuous on R, equipped with the uniform norm || · ||∞ . Chui and Li introduced the MRA’s and the corresponding wavelet structures in Cu (See [1]. The similar discussion can also be found in [3]. ) Let Nm (x) be the m-th order B-spline defined recursively by N1 (x) = χ[0,1) (x) and Nm (x) = Nm−1 ∗ N1 (x) = R1 Nm−1 (x − t) dt. For convenience, we write ψ0 (x) = N2m (x). It is known that 0 ψ0 satisfies the following two-scaling equation: ¶ 2m µ X 2m −(2m−1) ψ0 (x) = 2 ψ0 (2x − k). (1) k k=0

By setting

X

Vj = {

ck ψ0 (2j x − k);

(ck ) ∈ `∞ },

k∈Z

the authors of [1] point out that {Vj }j∈Z forms an MRA of Cu in the following S T sense: Cu = Vj , Vj = R. Then the wavelet function corresponding to this MRA can be defined by X ψ1 (x) = (−1)k−1 ψ0 (k − 1)ψ0 (2x − k). (2) k∈Z

P Letting P0 (z) = Ψ0 (z) = 12 k∈Z ψ0 (k)z k , and Q0 (z) = zΨ0 (−z), we can rewrite equations (1.1) and (1.2) to ω ψb0 (ω) = P0 (z)ψb0 ( ), 2 ω z = e−iω/2 , ψb1 (ω) = Q0 (z)ψb0 ( ), 2 2m ( 1+z , 2 )

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R where fb is the Fourier transform of f : fb(ω) := R f (x)e−ixω dx. Now we define the wavelet subspace X Wj := { dk ψ1 (2j x − k), (dk ) ∈ l∞ }, k∈Z

L L which is a complement of Vj in Vj+1 : W Vj = Vj+1 , where denotes the Tj L direct sum, i.e. Wj + Vj = Vj+1 and Vj Wj = ∅. Therefore Cu = R j∈Z Wj . It can be verify that g(2−j Z) = 0,

∀g ∈ Wj .

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Noting that Ψ0 (z) : Ψ1 (z) :

1X ψ0 (k)z k 6= 0, 2 k∈Z 1X = ψ1 (k + 1/2)z k 6= 0, 2

=

|z| = 1,

k∈Z

then ∀f ∈ Cu , we can define interpolating operators I0j : Cu → Vj such that I0j f (2−j Z) = f (2−j Z), and I1j : Cu → Wj such that I1j f (2−j (Z + 1/2)) = f (2−j (Z + 1/2)). By assertion (1.4), we have M j I1 , j ∈ Z. (4) I0j+1 = I0j L L where is theL“direct sum” L of operators defined by P1L P2 = P1 +P2 −P2 P1 . Generally, P1 P2 6= P2 P1 . Hence the notation α∈Λ Pα is only applied for the ordered set Λ, meanwhile the “sum” takes the same order. According to (1.5), decomposing a function into its wavelet components can be performed by the interpolating scheme. Hence we call ψ1 an interpolating wavelet. The aim of this paper is to construct interpolating wavelet packets for this structure.

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Construction of Wavelet Packets

S Let the set of natural numbers and Z+ = N {0}. We define Pn N denote Pn αd =j −j−1 d 2 for an integer d ∈ N whose dyadic expansion is d = j=0 j j=0 dj 2 with dn 6= 0 and set α0 = 0. It is clear that α2d = αd /2 and α2d+1 = (αd + 1)/2. In this paper, Id,j , (d, j) ∈ Z+ × Z, always denotes a dyadic interval [d2j , (d + 1)2j ). We also write I0,∞ = [0, ∞) . A set of dyadic intervals T = {Id,j }(d,j)∈ΓT is called a dyadic decomposition of an interval L ⊂ R if the intervals in T are S mutually disjoint and (d,j)∈ΓT Id,j = L. ( A dyadic interval may has (infinity) many dyadic decompositions.) Since the intervals in a dyadic decomposition T are mutually disjoint , we can order their indexes by setting (d, j) < (d0 , j 0 ) if 2

0

d2j < d0 2j . Later we always assume that the index set ΓT for T is well-ordered in this way. Now we are to construct the interpolating wavelet packet. Recall that functions ψ0 and ψ1 are already defined by (1.1) and (1.2) respectively. Now for any positive integer d ∈ N, we inductively define X ψ2d (x) = (−1)k ψd−1 (k + αd−1 )ψd (2x − k), k∈Z

ψ2d+1 (x) =

X

(−1)k−1 ψd (k + αd − 1)ψd (2x − k).

k∈Z

Similarly, by setting Ψd (z) =

1X ψd (k + αd )z k , 2

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k∈Z

and Pd (z) = Qd (z) =

Ψd−1 (−z) zΨd (−z)

we can rewrite (2.1) as following. ψb2d (ω) = ψb2d+1 (ω) =

ω Pd (z)ψbd ( ), 2 ω b Qd (z)ψd ( ), 2

z = e−iω/2 .

It can be verify that Ψ2d (z 2 ) = Ψ2d+1 (z 2 ) =

Ψd (−z)Ψd−1 (z) + Ψd (z)Ψd−1 (−z), 2Ψd (z)Ψd (−z).

We shall establish some properties for Pd (z) and Qd (z). For this purpose we introduce the notion of P´ olya frequency polynomials (PF polynomials). A polynomial is called a PF polynomial if its coefficients is a P´olya frequency sequence (cf. [4]). If the sequence is also symmetric, we call the corresponding polynomial an SPF polynomial. It is well-known that an SPF polynomial p(z) can be factored as n Y p(z) = z k (z − rj ), j=1

where each rj < 0 and rj rn−j = 1. If p(−1) 6= 0, 1 ≤ j ≤ n (which implies that n is even), then p(z) 6= 0 on the unit circle in the complex plane. Lemma 2.1 Let Gd (z) = Ψd ((−1)d z). Then Gd (z) is an SPF polynomial of degree 2m − 1 and Gd (−1) 6= 0.

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Proof. We use the mathematical induction to prove it. At first, G0 (z) = 2m (z) Ψ0 (z) = zΠ (2m−1)! , where Π2m (z) is the Euler-Frobenius Polynomial of degree 2m − 2 (See [2], [5]). Hence Ψ0 (z) is an SPF polynomial with Ψ0 (−1) 6= 0. The statement is true for d = 0. Let r0,j , 1 ≤ j ≤ 2m − 2, be all roots of G0 (z). By Q2m−2 2 identity (2.5), G1 (z) = 2z j=1 (z + r0,j ) which implies that the statement is also true for d = 1. Now for any d ∈ N we assume that the statement is already true for any l ∈ Z+ , l < d. We will prove it also true for d. In case of d = 2n + 1, we have Ψd (z 2 ) = 2Ψn (−z)Ψn (z). Since n < d, by the inductive assumption, Gn (z) in an SPF polynomial with Gn (−1) 6= 0. Similar to the proof for d = 1, we can claim the statement is true for d. Now consider the case of d = 2n. Setting Hn (z) = Gn (z)Gn−1 (z) := P2m−2 P2m−3 P4m−4 z 2 j=0 hj z j , Hne (z) = z j=0 h2j z j , and Hno (z) = z j=0 h2j+1 z j , we have Hn (z) = Hne (z 2 ) + zHno (z 2 ) and Gd (z) = 2Hne (z). Since Hn (z) is an SFP polynomial, so does Gd (z). Note that Hno (−1) = 0, then Gd (−1) = 2Hne (−1) = 2(Hn (i) − iHno (−1)) = 2Hn (i) 6= 0. Hence the statement is true for any d ∈ Z+ . Lemma 2.2 For any integer d ∈ N, we have the following. (1) supp ψd ⊂ [1, 2m]; (2) Ψd (z) 6= 0, |z| = 1; (3) ψd (Z + αn ) = 0, n ∈ Z+ , n < d. Proof. Since supp ψ0 ⊂ [0, 2m], by (1.2), supp ψ1 ⊂ [1, 2m]. Using the mathematical induction we can verify that, for any d ∈ N, the degree of Ψd (z) is 2m − 1 and z is a simple factor of Ψd (z). Therefore by (2.1), for any d ∈ N, suppψd ⊂ [1, 2m]. Statement (2) is a direct consequence of Lemma 1. Now we use the mathematical induction to prove statement (3). It is clear that statement (3) is true for d = 1. We will prove it also true for any d ∈ N under the assumption that it is true for any l, l < d. For this purpose, at first we point out the following fact. If for integers i ∈ Z+ and n ∈ N, ψn (Z + αi ) = 0, then for any k ∈ Z, ψn (2(Z + α2i ) − k) = 0 and ψn (2(Z + α2i+1 ) − k) = 0. Now we consider the case of d = 2n. By assertion (2.5), the inductive assumption, and the fact we just mentioned, we can obtain ψd (Z + αl ) = 0, 0 ≤ l < d. The statement is proven. In the case of d = 2n + 1, we can obtain ψd (Z + αl ) = 0, 0 ≤ l < d − 1 in the same way. Then we only need to verify ψd (Z + αd−1 ) = 0. It can be derived from the calculation of X ψ(l + α2n ) = (−1)k−1 ψn (k − l) + αn )ψ(2l + 2α2n − k) k∈Z

=

X

(−1)k−1 ψn (k − 1 + αn )ψ(2l + αn − k) = 0, ∀l ∈ Z.

k∈Z

From Lemma 1 and Lemma 2 we obtain the following. P Lemma 2.3 Let Wjd = { k∈Z ck ψd (2j x−k); (ck ) ∈ `∞ } and Idj be the interpolating operator from Cu to Wjd such that Idj f (2−j (Z + αd )) = f (2−j (Z + αd )). Then for any d ∈ Z+ and j ∈ Z, we have 4

0

(1) If Id,j ⊂ Id0 ,j 0 , then Wjd ⊂ Wjd0 ; L 2d+1 2d Wj−1 ; (2) Wjd = Wj−1 j−1 L j−1 (3) Idj = I2d I2d+1 . From Lemma 3, we can prove the following theorem. Theorem 2.4 Let T c be a dyadic decomposition of [0, ∞), T o a dyadic composition of (0, ∞), and T a dyadic decomposition of ID,J . Then the following statements hold.L (1) Cu = R (d,j)∈Γ 0 Wjd ; T L (2) Cu = (d,j)∈ΓT c Wjd ; L (3) WJD = (d,j)∈ΓT Wjd ; L J (4) ID = (d,j)∈ΓT Idj , provided that set ΓT is finite. Since ψd has the properties established in Theorem 1, we call W := {ψd }d∈N an interpolating wavelet packet. We do not add ψ0 into the packet, because ψ0 is not a wavelet, but a scaling function.

3

Dual Wavelet Packet

In order to form the dual wavelet packet of W, we introduce the inner product space L2(−m) = {f ; f (i) ∈ L2 , i = 0, 1, · · · , m}, R which is equipped with the inner product < f, g >:= f (m) (x)g (m) (x) dx. We write f ⊥ g, if < f, g >= 0. It is obvious that W ⊂ L2(−m) . Let Wjd,2 = {

X

ck ψ(2j x − k),

(ck ) ∈ `2 }.

k∈Z

We have the following. T

Lemma 3.1 If Id,j

0

Id0 ,j 0 = ∅, then Wjd,2 ⊥ Wjd0 ,2 .

Proof. Write f [2m−1] (x) = f (2m−1) (x+) − f (2m−1) (x−). By using the mathematical induction we can prove that ψd∗ (l + αd ) = 22+(2m−1) ψd−1 (l + αd−1 ), where [x] is the integer part of x. By assertion (3.1), we obtain ∗ ψ2d (l + α2d+1 ) X 2m−1 (−1)k ψd−1 (k + αd−1 )ψd∗ (2l + αd + 1 − k) = 2 k∈Z

= 2

d+3 2 (2m−1)

X

(−1)k ψd−1 (k + αd−1 )ψd−1 (2l + 1 − k + αd−1 ) = 0.

k∈Z

5

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It follows that Sd := {x; from (3.1) and Lemma 1,

ψd∗ (x) 6= 0} ⊂

< ψd (x − k), ψd0 (x − k 0 ) >= (−1)m

Sd

n=0 (Z

X

+ αn ). Note that if d < d0 ,

ψd∗ (x)ψd0 (x + (k − k 0 )) = 0,

x∈Sd 0

then Wjd,2 ⊥ WjdT,2 . The lemma is true in the case of j = j 0 . In the case of j < j 0 , since Id,j Id0 ,j 0 = ∅. there is a d1 , d1 6= d0 , such that Id,j ⊂ Id1 ,j 0 , and 0 0 therefore Wjd,2 ⊂ Wjd01 ,2 . Since Wjd01 ,2 ⊥ Wjd0 ,2 , Wjd,2 ⊥ Wjd0 ,2 . In the case of j > j 0 , the proof is similar. ˜ := {ψ˜d }d∈N ⊂ L2 Definition 3.1 W (−m) is called a dual wavelet packet of W(with respect to L2(−m) ) if 0

< ψd (2j x − k), ψ˜d0 (2j x − k 0 ) >= 0 for Id,j

T

Id0 ,j 0 = ∅, and < ψ˜d , ψd >= 1.

By lemma 4, if ψ˜d is chosen from W0d , then the first condition in Definition ˜ := {ψ˜d }d∈N will be 1 is automatically satisfied for j 6= j0. In this case, W a dual wavelet packet of W provided that {ψ˜d (x − k)}k∈Z is a dual basis of {ψd (x − k)}k∈Z in space W0d for each d ∈ N. P Lemma 3.2 Let Ψd (z) = k∈Z ψd (k + αd )z k . then Ψd (z) 6= 0 on |z| = 1. Proof. From (3.1), we have Ψd (z) = 2[

d+1 2 ](2m−1)

Ψd−1 (z).

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Hence the lemma follows Lemma 1. Theorem 3.3 Let Gd (z) = (−1)m [Ψd (z)Ψd (z)]−1 = X ψ˜d (x) = gk ψd (x − k).

P

k∈Z gk z

k

and set (8)

k∈Z

˜ = {ψ˜d }d∈N is a dual wavelet packet of W (with respect to L2 Then W (−m) ). Proof. Since Ψd (z)Ψd (z) 6= 0 on the unit circle |z| = 1, the sequence {gk } is exponential decay as |k| → ∞, and therefore ψ˜d ∈ L2(−m) . Defining ˜ d (z) = P ˜ d (z) = Gd (z)Ψd (z). Then Ψ ψ˜d (k + αd )z k , we have Ψ k∈Z

X

< ψd (x), ψ˜d (x − l) > z l

˜ d (z) = (−1)m Ψ(z)Ψ

l∈Z

= Ψd (z)[Ψd (z)Ψd (z)]−1 Ψd (z) = 1. The theorem is proven. 6

Now we turn back to the space Cu . Let the dual space of Cu be denoted by Cu∗ , which contains all linear bounded functionals on Cu . For f ∗ ∈ Cu∗ and f ∈ Cu , the functional value of f ∗ at f is denoted by (f, f ∗ ). It is in Cu∗ that the classical Dirac delta function (distribution) δ defined by 2m (f, δ) = f (0), f ∈ Cu . It can be verified that dd2m x ψ˜d (x) is also a functional in Cu∗ . In fact, it is a linear combination of the Dirac delta function and its integer translates with exponential decay coefficients. Definition 3.2 W ∗ := {ψd∗ }d∈N ⊂ Cu∗ is called a dual wavelet packet of W 0 (with respect to Cu ) if (ψd (2j x − k), ψd∗0 (2j x − k 0 )) = 0 for Id,j ∩ Id0 ,j 0 = ∅, and (ψd , ψd∗ ) = 1. Now we give a dual wavelet packet of W with respect of Cu . P k Theorem 3.4 Let (−1)m Ψ−1 k∈Z βk z , and set d (z) := X ψd∗ (x) := βk δ(x − k).

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k∈Z

Then W ∗ = {ψd∗ }d∈N is a dual wavelet packet of W (with respect to Cu ). Proof. We have 0 0 < ψd (2 x), ψ˜d (2j (x − l)) >= (−1)m 22mj

j

and

µ ¶ d2m ˜ j 0 j ψd (2 x), 2m ψd (2 (x − l)) d x

µ ¶ 0 0 d2m (ψd (2j x), ψd∗ (2j (x − l))) = (−1)m ψd (2j x), 2m ψ˜d (2j (x − l)) . d x

It follows that 0 0 0 (ψd (2j x), ψd∗ (2j (x − l)) = 2−2mj < ψd (2j x), ψ˜d (2j (x − l)) > .

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Then the theorem can be derived from (3.5) and Theorem 2.

References [1] Chui, C. K. and C. Li, Dyadic affine Decompositions and functional wavelet transforms, SJMA, 1994, to appear. [2] Chui, C., K. and J. Z. Wang, On compactly supported wavelet and a duality principle, TAMS 330 (1992), 903–915. [3] Donoho, D., L., Interpolating wavelet transform, 1992, preprint. [4] Karlin, S., Total Positivity, Stanford University Press, Stanford, California, 1968. [5] Scheoberg, I., J., Cardinal Spline Interpolation, CBMS-NSF Series in Applied Math. #12, SIAM Pub., Philadephia, 1973.

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