JAMJournal of Applied Mathematics1687-00421110-757XHindawi Publishing Corporation84685210.1155/2012/846852846852Research ArticleOrthogonal Multiwavelet Frames in L2RdZhanweiLiu1GuoenHu2GuochangWu3XuYuesheng1School of Information EngineeringZhengzhou UniversityZhengzhou 450001Chinazzu.edu.cn2Department of Applied MathematicsZhengzhou Information Science and Technology InstituteZhengzhou 450002China3Department of Applied MathematicsHenan University of Economics and LawZhengzhou 450002Chinainchina.cc20121512201120122706201121092011251020112012Copyright © 2012 Liu Zhanwei et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We characterize the orthogonal frames and orthogonal multiwavelet frames in L2Rd with matrix dilations of the form (Df)(x)=detAf(Ax), where A is an arbitrary expanding d×d matrix with integer coefficients. Firstly, through two arbitrarily multiwavelet frames, we give a simple construction of a pair of orthogonal multiwavelet frames. Then, by using the unitary extension principle, we present an algorithm for the construction of arbitrarily many orthogonal multiwavelet tight frames. Finally, we give a general construction algorithm for orthogonal multiwavelet tight frames from a scaling function.

1. Introduction

Wavelets are mathematical functions that take account into the resolutions and the frequencies simultaneously . Moreover, wavelets could cut up data into different frequency components such that people can study each component with a resolution matched to its scale.

The classical MRA scaler wavelets are probably the most important class of orthonormal wavelets. However, the scalar wavelets cannot have the orthogonality, compact support, and symmetry at the same time (except the Haar wavelet). It is a disadvantage for signal processing. Multiwavelets have attracted much attention in the research community, since multiwavelets have more desired properties than any scalar wavelet function, such as orthogonality, short compact support, symmetry, and high approximation order . It is natural, therefore, to develop the multiwavelets theory that can produce systems having these properties.

Although many compression applications of wavelets use wavelet or multiwavelet bases, the redundant representation offered by wavelet frames has already been put to good use for signal denoising and image compression. In fact, the concept of frame was introduced a long time ago  and has received much attention recently due to the development and study of wavelet theory [9, 10]. In particular, inspired by these and other applications, many people are interested in some types of frames, such as tight wavelet frames, dual wavelet frames, and orthogonal frames .

In , Weber proposed orthogonal wavelet frames, which are useful in multiple access communication systems and superframes. Later in , authors discussed a pair of orthogonal frames to be orthogonal in a shift-invariant space. In , authors presented sufficient conditions for the construction of orthogonal MRA wavelet frames in L2(R). This led them to a vector-valued discrete wavelet transform. But all these results just base on 2 dilation wavelet transform. In this paper, we present the construction of orthogonal multiwavelet frames in L2(Rd) with matrix dilation, where the basic ingredients consists of two fixed multiwavelet basis and a paraunitary matrix of an appropriate size. Furthermore, by using the unitary extension principle, we present an algorithm for the construction of orthogonal multiwavelet tight frames from two suitable functions and give a general construction algorithm for orthogonal multiwavelet tight frames from a scaling function. These constructions lead to filter banks in l2(Zd) with similar orthogonality relations.

Let us now describe the organization of the material that follows. Section 2 contains some definitions in this paper. Also, we review some relative notations. In Section 3, we describe the construction of orthogonal multiwavelet frames and present different algorithms for the construction of orthogonal multiwavelet tight frames in L2(Rd) with matrix dilation.

2. Preliminaries

Let us now establish some basic notations.

We denote by Td the d-dimensional torus. By L2(Td), we denote the space of all Zd-periodic functions f (i.e., f is 1-periodic in each variable) such that Td|f(x)|2dx<+. The subsets of Rd invariant under Zd translations and the subsets of Td are often identified.

We use the Fourier transform in the formf̂(ω)=Rdf(x)e-2πix,ωdx, where ·,· denotes the standard inner product in Rd. The Fourier inverse transform is defined byf(x)=f̂(ω)̌=Rdf̂(ω)e2πix,ωdω.

Let Ed denote the set of all expanding d×d matrices A with integer coefficients. The expanding matrices mean that all eigenvalues have magnitude greater than 1. For AEd, we denote by B the transpose of A. It is obvious that BEd.

A collection of elements {ϕj:jJ} in a Hilbert space H is called a frame if there exist constants a and b, 0<ab<, such thataf2jJ|f,ϕj|2bf2,  for  all  fH. If {ϕj:jJ} satisfies the second inequality, then {ϕj:jJ} is called a Bessel sequence. Let a0 the supremum of all such numbers a and b0 the infimum of all such numbers b, then a0 and b0 are called the frame bounds of the frame {ϕj:jJ}. When a0=b0, we say that the frame is tight. When a0=b0=1, we say the frame is a Parseval frame.

In this paper, we will work with two families of unitary operators on L2(Rd). The first one consists of all translation operators Tk:L2(Rd)L2(Rd),  kZd, defined by (Tkf)(x)=f(x-k). The second one consists of all integer powers of the dilation operator DA:L2(Rd)L2(Rd) defined by (Df)(x)=|A|f(Ax) with AEd.

Let us now fix an arbitrary matrix AEd. For Ψ={ψ1,,ψr}L2(Rd), we will consider the affine system X(Ψ) defined byX(Ψ)={ψj,  kl(x)ψj,kl(x)=|detA|j/2ψl(Ajx-k):jZ;  kZd;  l=1,,r}.

Then, we define the multiwavelet frame, the multiwavelet tight frame, the multiwavelet tight frame, and the filter.

Definition 2.1.

We say that X(Ψ)L2(Rd) is a multiwavelet frame if the system (2.4) is a frame for L2(Rd).

Definition 2.2.

We say that X(Ψ)L2(Rd) is a multiwavelet tight frame if the system (2.4) is a tight frame for L2(Rd).

Definition 2.3.

We say that X(Ψ)L2(Rd) is a multiwavelet tight frame if the system (2.4) is a Parseval frame for L2(Rd).

We turn to the concept of multiresolution analysis (MRA) in L2(Rd) which is a useful tool in our study.

Definition 2.4.

Let {Vm}mZ be a sequence of closed subspaces of L2(Rd) satisfying:

VjVj+1,

jZ¯Vj=L2(Rd),

jZVj={0},

f(x)Vjf(Ax)Vj+1,  jZ, where AEd,

There exists a function ϕ(x)V0 such that {ϕ(x-k)}kZd is a frame of V0.

Then, {Vj}jZ is called an MRA and the function ϕ in (5) a scaling function.

There is a standard procedure for constructing multiwavelets from a given MRA(Vj). Firstly, one defines Wj=Vj+1Vj for all jZ. As an easy consequence of conditions (1)–(4) from Definition 2.4, one obtains L2(Rd)=jZWj¯ and Wj+1=DWj, for all jZ. Suppose now that there exist functions ΨW0 such that the system E(Ψ)={ψ(·-k):kZd,ψΨ} is a frame for W0. Then, {DjTkψ:kZd,ψΨ} is a frame for Wj, for all jZ, and, consequently, {DjTkψ:jZ,kZd,ψΨ} is a frame for L2(Rd).

In the following, we will borrow some notations from [17, 18] which will be used in this paper.

Let X be a (countable) Bessel system for a separable Hilbert space H over the complex field C. The synthesis operator TX:l2(X)H, which is well known to be bounded, is defined by TXa=hXahh for a={ah}hX. The adjoint operator TX* of TX, called the analysis operator, is TX*:Hl2(X);  TX*f={f,h}hX. Recall that X is a frame for H if and only SX=TXTX*:HH, the frame operator or dual Gramian, is bounded and has a bounded inverse [20, 21], and it is a tight frame (with frame bound 1) if and only if SX is the identity operator. The system X is a Riesz system (for span¯X) if and only its Gramian GX=TX*TX is bounded and has a bounded inverse; it is an orthonormal system of H if and only if GX is the identity operator.

Definition 2.5.

Let X and Y=RX, where R:hRh is a bijection between X and Y, be two frame for H. We call X and Y a dual frames for H if TYTX*=I, that is, hXf,hRh=f for all fH.

Definition 2.6.

Let X and Y=RX, where R:hRh is a bijection between X and Y, be two frames for H. We call X and Y a pair of orthogonal frames for H if TYTX*=0, that is, hXf,hRh=0 for all fH.

Definition 2.7.

A closed subspace VL2(Rd) is shift invariant if for  all  fV implies TkfV for any kZd.

We consider orthogonal frames in a shift-invariant subspace of L2(Rd). Let Φ be a countable subset of L2(Rd), and E(Φ)={ϕ(·-k):kZd,ϕΦ}. Define S(Φ)=span¯E(Φ), the smallest closed subspace that contains E(Φ). Throughout the rest of this paper, we assume that E(Φ) is a Bessel sequence for S(Φ). This assumption settles most of the convergence issues. The space S(Φ) is called the shift-invariant space generated by Φ and Φ a generating set for S(Φ). Shift-invariant spaces have been studied extensively in the literature, for example, [22, 23].

For ωRd, we define the pre-Gramian byJΦ(ω)=(ϕ̂(ω+α))αZd,ϕΦ, where ϕ̂ is the Fourier transform of ϕ. Note that the domain of the pre-Gramian matrix as an operator is l2(Φ) and its codomain is l2(Zd). The pre-Gramian can be regarded as the synthesis operator represented in Fourier domain as it was extensively studied in .

Let Φ and Ψ=RΦ, where R is a bijection satisfying R(ϕ(·-k))=(Rϕ)(·-k), be countable subsets of L2(Rd). Suppose that S(Φ)=S(Ψ) and that both E(Φ) and E(Ψ) are frames for S(Φ). Then, by definition, E(Φ) and E(Ψ) are a pair of orthogonal frames for S(Φ) if and only if for all fS(Φ),Sf=TE(Ψ)TE(Φ)*=0.

We define the mixed dual Gramian as G̃(ω)=JΨ(ω)JΦ*(ω) and Gramians asGΦ(ω)=JΦ*(ω)JΦ(ω),GΨ(ω)=JΨ*(ω)JΨ(ω). Then, it is proven in  that, for any fL2(Rd),(Sf̂)|ω+α=G̃(ω)f̂|ω+α, where f̂|ω+α is the column vector (f̂(ω+α))αZdT. By (2.8), one can prove easily that Sf=0 for all fL2(Rd) if and only if G̃(ω)=0 for a.e. ωRd.

3. Orthogonal Multiwavelet Frames

In this section, we present a simple construction of a pair of orthogonal multiwavelet frames from two arbitrarily multiwavelet frames and get some interesting properties about the orthogonal multiwavelet frames. We also show different algorithms for the construction of arbitrarily many orthogonal multiwavelet tight frames.

Firstly, we give a lemma, which has been obtained by Weber in .

Lemma 3.1.

Let Ψ1={ψ11,ψ21,,ψr1} and Ψ2={ψ12,ψ22,,ψr2}. Suppose that X(Ψ1) and X(Ψ2) are multiwavelet frames for L2(Rd). X(Ψ1) and X(Ψ2) are a pair of orthogonal frames for L2(Rd) if and only if the following two equations are satisfied a.e.:   i=1rjZψi1̂(Bjω)ψi2̂(Bjω)¯=0,  a.e.,  i=1rj=0+ψi1̂(Bjω)ψi2̂(Bj(ω+q))¯=0,a.e.,  kZd,  qZdBZd.

From Lemma 3.1, by Theorem 2.3 , we can construct a pair of orthogonal multiwavelet frames easily.

Theorem 3.2.

Let Ψ1={ψ11,ψ21,,ψr1}  and  Ψ2={ψ12,ψ22,,ψr2}L2(Rd) for some positive integer r. Suppose that X(Ψ1) and X(Ψ2) are multiwavelet frames for L2(Rd). Let V=(V1;V2) be a 2r×2r constant unitary matrix, where V1 is the submatrix of the first r columns and V2 the remaining r columns. Then, X(Ψ11) and X(Ψ22) are a pair of orthogonal multiwavelet frames for L2(Rd), where Ψ11=V1Ψ1 and Ψ22=V2Ψ2.

Proof.

Assume that V is a constant matrix such that Ψ̂11=V1Ψ̂1 and Ψ̂22=V2Ψ̂2. Then, one can directly calculate the dual Gramians of Xq(Ψ11) and Xq(Ψ22). It follows from the fact that the double sums in (3.1) are the entries of the dual Gramian of the affine systems .

Let V=(vlm)1l,m2r. For a fixed qZdBZd, i1,2, we have l=12rm0ψliî¯(Bmω)ψliî(Bm(ω+q))=l=12rm0n=1rvl,n¯ψnî¯(Bmω)n=1rvl,nψnî(Bm(ω+q))=m0n=1rψnî¯(Bmω)n=1rψnî(Bm(ω+q))l=12rvl,n¯vl,n=m0n=1rψnî¯(Bmω)ψnî(Bm(ω+q)), where we used the fact that the double sums converge absolutely a.e., V*V=I2r, and that X(Ψ1) and X(Ψ2) are frames for L2(Rd). Moreover, l=12rmZψliî¯(Bmω)ψliî(Bmω)=l=12rmZn=1rvl,n¯ψnî¯(Bmω)n=1rvl,nψnî(Bmω)=mZn=1rψnî¯(Bmω)n=1rψnî(Bmω)l=12rvl,n¯vl,n=mZn=1rψnî¯(Bmω)ψnî(Bmω). From the above results, by using the dual Gramian characterization of frames in [25, Corollary 5.7], then X(Ψ11) and X(Ψ22) are frames for L2(Rd).

We now show that the multiwavelet systems generated by Ψ11 and Ψ22 are a pair of orthogonal frames for L2(Rd). We apply Lemma 3.1 to Ψ11:={ψ111,ψ211,,ψ2r11} and Ψ22:={ψ122,ψ222,,ψ2r22}. Let V=(vlm)1l,m2r. For all qZdBZd, we have l=12rm0ψl11̂¯(Bmω)ψl22̂(Bm(ω+q))=l=12rm0n=1rvl,n¯ψn1̂¯(Bmω)n=1rvl,r+nψn2̂(Bm(ω+q))=m0n=1rψn1̂¯(Bmω)n=1rψn2̂(Bm(ω+q))l=12rvl,n¯vl,r+n=m0n=1rψn1̂¯(Bmω)n=1rψn2̂(Bm(ω+q))×0=0, where we used the orthogonality of the columns of V.

Moreover, l=12rmZψl11̂¯(Bmω)ψl22̂(Bm(ω))=l=12rmZn=1rvl,n¯ψn1̂¯(Bmω)n=1rvl,r+nψn2̂(Bm(ω))=mZn=1rψn1̂¯(Bmω)n=1rψn2̂(Bm(ω))l=12rvl,m¯vl,r+n=mZn=1rψn1̂¯(Bmω)n=1rψn2̂(Bm(ω))×0=0, by Lemma 3.1, Ψ11 and Ψ22 generate a pair of orthogonal frames.

The following results give some properties of the orthogonal frames.

Proposition 3.3.

Suppose that E(ψi) and E(ψj) are a pair of orthogonal affine Bessel sequences in L2(Rd). If αL2(Rd) is a Zd-periodic function, then E(ψi) and E(αψj) are a pair of orthogonal affine Bessel sequences.

Proof.

Suppose that E(ψi) and E(ψj) are a pair of orthogonal affine Bessel sequences in L2(Rd). Then, for all fL2(Rd), we have Sf(x)=mZdf(x),ψi(x+m)ψj(x+m)=0. Let ψj:=αψj. Since α is a Zd-periodic function, then E(ψj) is an affine Bessel sequence for L2(Rd) from the fact that, for all fL2(Rd), kZd|f(x),α(x-k)ψj(x-k)|2=kZd|α(x)¯f(x),ψj(x-k)|2Bα¯f2Bα¯2f2=Bf2.

Again by α being a Zd-periodic function, we have the following equation: Sf(x)=mZdf(x),ψi(x+m)ψj(x+m)=mZdf(x),ψi(x+m)ψj(x+m)α(x+m)=α(x)mZdf(x),ψi(x+m)ψj(x+m)=0. Hence, E(ψi) and E(αψj) are a pair of orthogonal affine Bessel sequences in L2(Rd).

Proposition 3.4.

Suppose that E(ψi) and E(ψj) are a pair of orthogonal frames for HL2(Rd). Let αL2(Rd) be a Zd-periodic function. If E(αψj) is a frame for H, then E(ψi) and E(αψj) are a pair of orthogonal frames for H.

Proof.

Similar to the proof in Proposition 3.3, we have the desired result.

Then, we recall a result from  that characterizes unitary extension principle (UEP) associated with more general matrix dilations in L2(Rd).

Lemma 3.5.

Suppose Φ=(ϕj)jJ is a refinable vector with a mask Γ such that jJφj2=RdΦ̂(ξ)l2(J)2dξ<,limjΦ̂(B-jξ)=1,for  a.e.  ξRd. Suppose also that Ψ=(ψj)jJ̃, where J̃={1,,N} is finite, is given by Ψ̂(Bξ)=H(ξ)Φ̂(ξ), where H=(hi,j)iJ̃,  iJ is a Zd-periodic, measurable matrix function satisfying Γ*(ξ)Γ(ξ+d)+H*(ξ)H(ξ+d)=Ω(ξ)δ0,d,for  a.e.  ξ, and for any dΥ, where Υ consists of representatives of distinct cosets of B-1Zd/Zd, then ΨL2(Rd) is a multiwavelet tight frame.

We call m a filter if mL([0,1)d). We shall call m a low-pass filter if m(0)=1, and we shall call m a high-pass filter if m(0)=0. Though not necessary, we will assume that every filter is continuous on a neighborhood of 0, so there will be no ambiguity in these definitions. Given a collection of filter M={m0,m1,,mr}L([0,1)d), let M(ξ) and M̃(ξ) be the matricesM(ξ)=(m0(ξ)m0(ξ+β)m1(ξ)m1(ξ+β)mr(ξ)mr(ξ+β)),  M̃(ξ)=(m1(ξ)m1(ξ+β)m2(ξ)m2(ξ+β)mr(ξ)mr(ξ+β)), where βΥ. In the remainder of the paper, the filter banks will be composed of a single low-pass filter (with index 0) and a number of high-pass filters.

With the above definitions, we present an algorithm for the construction of arbitrarily many orthogonal multiwavelet tight frames.

Theorem 3.6.

Suppose that ϕ1,ϕ2L2(Rd) are refinable functions which satisfy the conditions of the unitary extension principe, and let m1(ξ),m2(ξ) be the associated low-pass filter. Let M={m0(ξ),m1(ξ),,mr(ξ)} and N={n0(ξ),n1(ξ),,nr(ξ)} be filter banks with m0(ξ)=m1(ξ),  n0(ξ)=m2(ξ). For all βΥ, suppose that the following matrix equations hold:

M*(ξ)M(ξ)=I2 for almost every ξ,

N*(ξ)N(ξ)=I2 for almost every ξ,

M̃*(ξ)Ñ(ξ)=0 for almost every ξ.

Let η̂k(Bξ)=nk(ξ)ϕ̂2(ξ) and ψ̂k(Bξ)=mk(ξ)ϕ̂1(ξ),  1kr. Then, {ψ1,,ψr} and {η1,,ηr} generate orthogonal multiwavelet tight frames.

Proof.

For Items (a) and (b), by Lemma 3.5, then {ψ1,,ψr} and {η1,,ηr} generate multiwavelet tight frames. We use the characterization equations of Lemma 3.1 to prove orthogonality.

Let us focus on k=1rjZψ̂k(Bjξ)η̂k(Bjξ)¯. For each k, by Hölder’s inequality and virtue of the fact that ψk and ηk generate Bessel sequences [4, Theorem 8.3.2], we have jZ|ψ̂k(Bjξ)η̂k(Bjξ)¯|jZ|ψ̂k(Bjξ)|2jZ|η̂k(Bjξ)|2<, then the order of summation can be reversed. With this, by Item (c), k=1rjZψ̂k(Bjξ)η̂k(Bjξ)¯=k=1rjZmk(Bjξ)ϕ̂1(Bjξ)nk(Bjξ)ϕ̂2(Bjξ)¯=jZϕ̂1(Bjξ)ϕ̂2(Bjξ)¯k=1rmk(Bjξ)nk(Bjξ)¯=0 holds for almost every ξ.

Likewise, for qZdBZd, by item (c), k=1rj=0ψ̂k(Bjξ)η̂k(Bj(ξ+q))¯=k=1rj=0mk(Bj-1ξ)ϕ̂1(Bj-1ξ)nk(Bj-1(ξ+q))ϕ̂2(Bj-1(ξ+q))¯=j=0ϕ̂1(Bjω)ϕ̂2(Bj(ω+B-1q))¯k=1rmk(Bjω)nk(Bjω+Bj-1q))¯=0, where ω=B-1ξ.

The following results show the relationship between a pair of orthogonal MRA multiwavelet frames.

Theorem 3.7.

Suppose that X(Ψi) and X(Ψj) are a pair of orthogonal MRA multiwavelet frames, where Ψi={ψ1i,ψ2i,,ψri}, Ψj={ψ1j,ψ2j,,ψrj}. If S(Ψi)=S(Ψj) and there exist functions p,wL2(Rd) such that ΨiP={ψ1ip,ψ2ip,,ψrip} and ΨjP={ψ1jw,ψ2jw,,ψrjw} are multiwavelet frames, where ψlp and ψlw defined by ψlip(ω)̂=ψ̂li(ω)p̂(ω), ψljw(ω)̂=ψ̂lj(ω)ŵ(ω),  1lr respectively, then X(Ψip) and X(Ψjw) are a pair of orthogonal multiwavelet frames for L2(Rd).

Proof.

Suppose that X(Ψi), X(Ψj) are a pair of orthogonal MRA multiwavelet frames and S(Ψi)=S(Ψj), then, by the property of MRA multiwavelet frames, for any nmZ, we have S(AmΨi)S(AnΨi). Hence, for all f1S(Ψi)0=Sf1(x)=l=1rkZdsZf1(x),ψli(Asx-k)ψlj(Asx-k)=l=1rkZdf1(x),ψli(x-k)ψlj(x-k).

For any fL2(Rd), define f=f1+f2, where f1S(Ψi),f2(L2(Rd)S(Ψi)), then, f1,f2=0. With this, we get Sf2(x)=l=1rkZdf2(x),ψli(x-k)ψlj(x-k)=0. Hence, for all fL2(Rd), the following equation holds: Sf(x)=l=1rkZdf(x),ψli(x-k)ψlj(x-k)=l=1rkZdf1(x),ψli(x-k)ψlj(x-k)+l=1rkZdf2(x),ψli(x-k)ψlj(x-k)=0.

Notice that Ψj:={ψ1j,,ψrj}, since ψljw(ξ)̂=ψ̂j(ξ)ŵ(ξ),1lr, by Sf(x)=0, then 0=Sf(x)̂=l=1rkZdf(x),ψli(x-k)ψlĵ(ω)e-2πikω=l=1rkZdf1(x),ψli(x-k)ψlĵ(ω)e-2πikω+l=1rkZdf2(x),ψli(x-k)ψlĵ(ω)e-2πikω=w(ω)(l=1rkZdf1(x),ψli(x-k)ψlĵ(ω)e-2πikω+l=1rkZdf2(x),ψli(x-k)ψlĵ(ω)e-2πikω)=l=1rkZdf1(x),ψli(x-k)ψljŵ(ω)e-2πikω+l=1rkZdf2(x),ψli(x-k)ψljŵ(ω)e-2πikω=l=1rkZdf(x),ψli(x-k)ψljŵ(ω)e-2πikω.

Applying Fourier inverse transform on (3.19), we have 0=Sf(x)̂̌=l=1rkZdf(x),ψli(x-k)ψljŵ̌(ω)e-2πikω=l=1rkZdf(x),ψli(x-k)ψljw(x-k).

From the above result, we get the following equation: l=1rkZdf(x),ψli(x-k)f(x),ψljw(x-k)=f(x),l=1rkZdf(x),ψli(x-k)ψljw(x-k)=f(x),0=f(x),l=1rkZdf(x),ψljw(x-k)ψli(x-k), hence, l=1rkZdf(x),ψljw(x-k)ψli(x-k)=0. Similar to the calculation of (3.19), clearly l=1rkZdf(x),ψljw(x-k)ψlip(x-k)=0.

For any sZ, l=1rkZdf(x),ψljw(Asx-k)ψlip(Asx-k)=A-sl=1rkZdf(A-sx),ψljw(x-k)ψlip(x-k). Let g(x):=f(A-sx). Define operator T:L2(Rd)L2(Rd);  Tf(x)=g(x), obviously T is a surjection operator. If s is fixed, for all gL2(Rd), we get l=1rkZdf(x),ψljw(Asx-k)ψlip(Asx-k)=A-s(l=1rkZdg(x),ψljw(x-k)ψlip(x-k))=0.

Putting everything together, we have l=1rsZkZdf(x),ψljw(Asx-k)ψlip(Asx-k)=0, then, X(Ψip) and X(Ψjw) are a pair of orthogonal multiwavelet frames.

The following theorem describes a general construction algorithm for orthogonal multiwavelet tight frames.

Theorem 3.8.

Suppose K(ξ) is an r×r paraunitary matrix with B-1Zd-periodic entries ak,s(ξ); let Kj(ξ) denote the jth column. For all βΥ, suppose M={m0(ξ),m1(ξ),,mr(ξ)} and M*(ξ)M(ξ)=I2 hold for almost every ξ, where m0 and {m1,,ml} are low- and high-pass filters, respectively, for a multiwavelet tight frame with scaling function ϕ. For j=1,,r, define new filters via (n1,1j(ξ)n1,rj(ξ)nl,1j(ξ)nl,rj(ξ))=(Kj(ξ)m1(ξ)Kj(ξ)ml(ξ)). Then, for j=1,,r, the affine systems generated by Ψj={ψi,tj:i=1,,l,t=1,,r} obtained via ψ̂i,tj(Bξ)=ni,tj(ξ)ϕ̂(ξ) are multiwavelet tight frames and are pairwise orthogonal.

Proof.

Firstly, we prove that X(Ψj),1jr, are multiwavelet tight frames. Assume Mj={m0(ξ),n1,1j(ξ),,n1,rj(ξ),,nl,1j(ξ),,nl,rj(ξ)}. Define Mj(ξ) according to (3.12): Mj(ξ)=(m0(ξ)m0(ξ+β)n1,1j(ξ)n1,1j(ξ+β)n1,rj(ξ)n1,rj(ξ+β)nl,1j(ξ)nl,1j(ξ+β)nl,rj(ξ)nl,rj(ξ+β)), where βΥ. Then, Mj*(ξ)Mj(ξ) is a 2  ×  2 matrix. Next, we examine the entries of Mj*(ξ)Mj(ξ) individually. Note that the columns of K(ξ) have length 1, by M*(ξ)M(ξ)=I2, it follows that [Mj*(ξ)Mj(ξ)]1,1=|m0(ξ)|2+k=1rt=1l|ak,j(ξ)mt(ξ)|2=|m0(ξ)|2+k=1r|ak,j(ξ)|2t=1l|mt(ξ)|2=|m0(ξ)|2+t=1l|mt(ξ)|2  =1, where [Mj*(ξ)Mj(ξ)]1,1 means the (1,1) entry of the matrix Mj*(ξ)Mj(ξ).

Similarly, [Mj*(ξ)Mj(ξ)]2,2=|m0(ξ+β)|2+k=1rt=1l|ak,j(ξ+β)mt(ξ+β)|2=|m0(ξ+β)|2+k=1r|ak,j(ξ+β)|2t=1l|mt(ξ+β)|2=|m0(ξ+β)|2+t=1l|mt(ξ+β)|2  =1.

Now, since the entries of K(ξ) are B-1Zd-periodic, again by M*(ξ)M(ξ)=I2, [Mj*(ξ)Mj(ξ)]1,2=m0(ξ+β)m0(ξ)¯+k=1rt=1lak,j(ξ)mt(ξ)¯ak,j(ξ+β)mt(ξ+β)=m0(ξ+β)m0(ξ)¯+k=1r|ak,j(ξ)|2t=1lmt(ξ)¯mt(ξ+β)=m0(ξ+β)m0(ξ)¯+t=1lmt(ξ)¯mt(ξ+β)=0. Finally, the (2,1)-entry must be zero by conjugate symmetry of Mj*(ξ)Mj(ξ). Hence, Mj*(ξ)Mj(ξ)=I2,1jr. Putting everything together, from Theorem 3.6, the affine systems generated by {ψi,tj:i=1,,l,t=1,,r} obtained via ψ̂i,tj(Bξ)=ni,tj(ξ)ϕ̂(ξ) are multiwavelet tight frames.

For orthogonality, according to (3.12), for j=1,,r, we have M̃j(ξ)=(n1,1j(ξ)n1,1j(ξ+β)n1,rj(ξ)n1,rj(ξ+β)nl,1j(ξ)nl,1j(ξ+β)nl,rj(ξ)nl,rj(ξ+β))=(Kj(ξ)m1(ξ)Kj(ξ+β)m1(ξ+β)Kj(ξ)ml(ξ)Kj(ξ+β)ml(ξ+β)). If 1jjr, then M̃j*(ξ)M̃j(ξ)=(Kj(ξ)m1(ξ)Kj(ξ+β)m1(ξ+β)Kj(ξ)ml(ξ)Kj(ξ+β)ml(ξ+β))*(Kj(ξ)m1(ξ)Kj(ξ+β)m1(ξ+β)Kj(ξ)ml(ξ)Kj(ξ+β)ml(ξ+β))=(Kj*(ξ)Kj(ξ)t=1l|mt(ξ)|2Kj*(ξ)Kj(ξ+β)t=1lmt(ξ)¯mt(ξ+β)Kj*(ξ+β)Kj(ξ)t=1lmt(ξ+β)¯mt(ξ)Kj*(ξ+β)Kj(ξ+β)t=1lmt(ξ+β)¯mt(ξ+β))=(Kj*(ξ)Kj(ξ)t=1l|mt(ξ)|2Kj*(ξ)Kj(ξ)t=1lmt(ξ)¯mt(ξ+β)Kj*(ξ)Kj(ξ)t=1lmt(ξ+β)¯mt(ξ)Kj*(ξ)Kj(ξ)t=1lmt(ξ+β)¯mt(ξ+β))=0, where we use the fact that the product of the two matrices Kj*(ξ)Kj(ξ) is 0 by the orthogonality of the columns of K(ξ). By Theorem 3.6, we have the desired result.

The following proposition is directly related to the construction algorithm in Theorem 3.8.

Proposition 3.9.

If ϕ is compactly supported, the paraunitary matrix K in Theorem 3.8 must have entries which are B-1Zd-periodic.

Proof.

The proof will follow the notation of Theorem 3.8. For 1jr, for all ξB-1Zd/Zd, the matrix Mj(ξ)=(m0(ξ)m0(ξ+β)a1,j(ξ)m1(ξ)a1,j(ξ+β)m1(ξ+β)ar,j(ξ)m1(ξ)ar,j(ξ+β)m1(ξ+β)a1,j(ξ)ml(ξ)a1,j(ξ+β)ml(ξ+β)ar,j(ξ)ml(ξ)ar,j(ξ+β)ml(ξ+β)) satisfies the equation Mj*(ξ)Mj(ξ)=I2a.e.ξ. Then, for almost every ξ, the following equation m0(ξ+β)m0(ξ)¯+k=1rak,j(ξ)¯ak,j(ξ+β)t=1lmt(ξ)¯mt(ξ+β)=0 must hold. Notice that m0 and {m1,,ml} are low- and high-pass filters, respectively, which meet Theorem 3.8. Then, m0(ξ+β)m0(ξ)¯+t=1lmt(ξ)¯mt(ξ+β)=0. Thus, we have m0(ξ+β)m0(ξ)¯=-t=1lmt(ξ)¯mt(ξ+β).

From the above results, we get the following equation: 0=m0(ξ+β)m0(ξ)¯+k=1rak,j(ξ)¯ak,j(ξ+β)t=1lmt(ξ)¯mt(ξ+β)=-t=1lmt(ξ)¯mt(ξ+β)+k=1rak,j(ξ)¯ak,j(ξ+β)t=1lmt(ξ)¯mt(ξ+β)=(k=1rak,j(ξ)¯ak,j(ξ+β)-1)t=1lmt(ξ)¯mt(ξ+β).

Hence, t=1lmt(ξ)¯mt(ξ+β)=0 or k=1rak,j(ξ)¯ak,j(ξ+β)=1. If ϕ is compactly supported, then the first possibility is eliminated except possibly on a set of measure 0, whence the second must hold almost everywhere. Now, the sum is precisely the inner product of the two vectors ak,j(ξ) and ak,j(ξ+β), each of which has length 1. Applying Cauchy-Schwarz inequation yields that the two vectors must be identical for almost every ξ.

4. Conclusion

In this paper, motivated by the notion of orthogonal frames, we present the construction of orthogonal multiwavelet frames in L2(Rd) with matrix dilation, where the basic ingredients consist of two fixed multiwavelet basis and a paraunitary matrix of an appropriate size. The number of orthogonal multiwavelet frames that can be constructed is arbitrary, and is determined by the size of the paraunitary matrix. Moreover, by using the unitary extension principle, we present an algorithm for the construction of orthogonal multiwavelet tight frames and give a general construction algorithm for orthogonal multiwavelet tight frames from a scaling function.

Acknowledgment

L. Zhanwei was supported by the Henan Provincial Natural Science Foundation of China (Grant no. 102300410205). H. Guoen was supported by the National Natural Science Foundation of China (Grant no. 10971228).

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