Research Article | Open Access
Orthogonal Multiwavelet Frames in
We characterize the orthogonal frames and orthogonal multiwavelet frames in with matrix dilations of the form , where is an arbitrary expanding 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.
Wavelets are mathematical functions that take account into the resolutions and the frequencies simultaneously [1–4]. 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 [5–7]. 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 [11–19].
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 . 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 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 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 with matrix dilation.
Let us now establish some basic notations.
We denote by the d-dimensional torus. By , we denote the space of all -periodic functions (i.e., is 1-periodic in each variable) such that . The subsets of invariant under translations and the subsets of are often identified.
We use the Fourier transform in the form where denotes the standard inner product in . The Fourier inverse transform is defined by
Let denote the set of all expanding matrices with integer coefficients. The expanding matrices mean that all eigenvalues have magnitude greater than 1. For , we denote by the transpose of . It is obvious that .
A collection of elements in a Hilbert space is called a frame if there exist constants and , , such that If satisfies the second inequality, then is called a Bessel sequence. Let the supremum of all such numbers and the infimum of all such numbers , then and are called the frame bounds of the frame . When , we say that the frame is tight. When , we say the frame is a Parseval frame.
In this paper, we will work with two families of unitary operators on . The first one consists of all translation operators , defined by . The second one consists of all integer powers of the dilation operator defined by with .
Let us now fix an arbitrary matrix . For , we will consider the affine system defined by
Then, we define the multiwavelet frame, the multiwavelet tight frame, the multiwavelet tight frame, and the filter.
Definition 2.1. We say that is a multiwavelet frame if the system (2.4) is a frame for .
Definition 2.2. We say that is a multiwavelet tight frame if the system (2.4) is a tight frame for .
Definition 2.3. We say that is a multiwavelet tight frame if the system (2.4) is a Parseval frame for .
We turn to the concept of multiresolution analysis (MRA) in which is a useful tool in our study.
Definition 2.4. Let be a sequence of closed subspaces of satisfying: ,, , , where ,There exists a function such that is a frame of . Then, is called an MRA and the function in (5) a scaling function.
There is a standard procedure for constructing multiwavelets from a given MRA. Firstly, one defines for all . As an easy consequence of conditions (1)–(4) from Definition 2.4, one obtains and , for all . Suppose now that there exist functions such that the system is a frame for . Then, is a frame for , for all , and, consequently, is a frame for .
Let be a (countable) Bessel system for a separable Hilbert space over the complex field . The synthesis operator , which is well known to be bounded, is defined by for . The adjoint operator of , called the analysis operator, is . Recall that is a frame for if and only , 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 is the identity operator. The system is a Riesz system (for ) if and only its Gramian is bounded and has a bounded inverse; it is an orthonormal system of if and only if is the identity operator.
Definition 2.5. Let and , where is a bijection between and , be two frame for . We call and a dual frames for if , that is, for all .
Definition 2.6. Let and , where is a bijection between and , be two frames for . We call and a pair of orthogonal frames for if , that is, for all .
Definition 2.7. A closed subspace is shift invariant if implies for any .
We consider orthogonal frames in a shift-invariant subspace of . Let be a countable subset of , and . Define , the smallest closed subspace that contains . Throughout the rest of this paper, we assume that is a Bessel sequence for . This assumption settles most of the convergence issues. The space is called the shift-invariant space generated by and a generating set for . Shift-invariant spaces have been studied extensively in the literature, for example, [22, 23].
For , we define the pre-Gramian by where is the Fourier transform of . Note that the domain of the pre-Gramian matrix as an operator is and its codomain is . The pre-Gramian can be regarded as the synthesis operator represented in Fourier domain as it was extensively studied in .
Let and , where is a bijection satisfying , be countable subsets of . Suppose that and that both and are frames for . Then, by definition, and are a pair of orthogonal frames for if and only if for all ,
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 and . Suppose that and are multiwavelet frames for . and are a pair of orthogonal frames for if and only if the following two equations are satisfied a.e.:
Theorem 3.2. Let for some positive integer r. Suppose that and are multiwavelet frames for . Let be a constant unitary matrix, where is the submatrix of the first r columns and the remaining r columns. Then, and are a pair of orthogonal multiwavelet frames for , where and .
Proof. Assume that is a constant matrix such that and . Then, one can directly calculate the dual Gramians of and . It follows from the fact that the double sums in (3.1) are the entries of the dual Gramian of the affine systems .
Let . For a fixed , , we have where we used the fact that the double sums converge absolutely a.e., , and that and are frames for . Moreover, From the above results, by using the dual Gramian characterization of frames in [25, Corollary 5.7], then and are frames for .
We now show that the multiwavelet systems generated by and are a pair of orthogonal frames for . We apply Lemma 3.1 to and . Let . For all , we have where we used the orthogonality of the columns of .
Moreover, by Lemma 3.1, and generate a pair of orthogonal frames.
The following results give some properties of the orthogonal frames.
Proposition 3.3. Suppose that and are a pair of orthogonal affine Bessel sequences in . If is a -periodic function, then and are a pair of orthogonal affine Bessel sequences.
Proof. Suppose that and are a pair of orthogonal affine Bessel sequences in . Then, for all , we have
Let . Since is a -periodic function, then is an affine Bessel sequence for from the fact that, for all ,
Again by being a -periodic function, we have the following equation: Hence, and are a pair of orthogonal affine Bessel sequences in .
Proposition 3.4. Suppose that and are a pair of orthogonal frames for . Let be a -periodic function. If is a frame for , then and are a pair of orthogonal frames for .
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 .
Lemma 3.5. Suppose is a refinable vector with a mask such that Suppose also that , where is finite, is given by where is a -periodic, measurable matrix function satisfying and for any , where consists of representatives of distinct cosets of , then is a multiwavelet tight frame.
We call a filter if . We shall call a low-pass filter if , and we shall call a high-pass filter if . 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 , let and be the matrices where . 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 are refinable functions which satisfy the conditions of the unitary extension principe, and let be the associated low-pass filter. Let and be filter banks with . For all , suppose that the following matrix equations hold: for almost every , for almost every , for almost every .Let and . Then, and generate orthogonal multiwavelet tight frames.
Proof. For Items (a) and (b), by Lemma 3.5, then and generate multiwavelet tight frames. We use the characterization equations of Lemma 3.1 to prove orthogonality.
Let us focus on . For each , by Hölder’s inequality and virtue of the fact that and generate Bessel sequences [4, Theorem 8.3.2], we have then the order of summation can be reversed. With this, by Item (c), holds for almost every .
Likewise, for , by item (c), where .
The following results show the relationship between a pair of orthogonal MRA multiwavelet frames.
Theorem 3.7. Suppose that and are a pair of orthogonal MRA multiwavelet frames, where , . If and there exist functions such that and are multiwavelet frames, where and defined by , respectively, then and are a pair of orthogonal multiwavelet frames for .
Proof. Suppose that , are a pair of orthogonal MRA multiwavelet frames and , then, by the property of MRA multiwavelet frames, for any , we have . Hence, for all
For any , define , where , then, . With this, we get Hence, for all , the following equation holds:
Notice that , since , by , then
Applying Fourier inverse transform on (3.19), we have
From the above result, we get the following equation: hence, Similar to the calculation of (3.19), clearly
For any , Let . Define operator , obviously is a surjection operator. If is fixed, for all , we get
Putting everything together, we have then, and are a pair of orthogonal multiwavelet frames.
The following theorem describes a general construction algorithm for orthogonal multiwavelet tight frames.
Theorem 3.8. Suppose is an paraunitary matrix with -periodic entries ; let denote the jth column. For all , suppose and hold for almost every , where and are low- and high-pass filters, respectively, for a multiwavelet tight frame with scaling function . For , define new filters via Then, for , the affine systems generated by obtained via are multiwavelet tight frames and are pairwise orthogonal.
Proof. Firstly, we prove that , are multiwavelet tight frames. Assume . Define according to (3.12):
where . Then, is a matrix. Next, we examine the entries of individually. Note that the columns of have length 1, by , it follows that
where means the (1,1) entry of the matrix .
Now, since the entries of are -periodic, again by , Finally, the (2,1)-entry must be zero by conjugate symmetry of . Hence, Putting everything together, from Theorem 3.6, the affine systems generated by obtained via are multiwavelet tight frames.
For orthogonality, according to (3.12), for , we have If , then where we use the fact that the product of the two matrices is 0 by the orthogonality of the columns of . 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 is compactly supported, the paraunitary matrix K in Theorem 3.8 must have entries which are -periodic.
Proof. The proof will follow the notation of Theorem 3.8. For , for all , the matrix
satisfies the equation
Then, for almost every , the following equation
must hold. Notice that and are low- and high-pass filters, respectively, which meet Theorem 3.8. Then,
Thus, we have
From the above results, we get the following equation:
Hence, or . If 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 and , each of which has length 1. Applying Cauchy-Schwarz inequation yields that the two vectors must be identical for almost every .
In this paper, motivated by the notion of orthogonal frames, we present the construction of orthogonal multiwavelet frames in 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.
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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