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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 896050, 10 pages
Minimum-Energy Bivariate Wavelet Frame with Arbitrary Dilation Matrix
School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan 750021, China
Received 31 January 2013; Revised 10 June 2013; Accepted 18 June 2013
Academic Editor: Li Weili
Copyright © 2013 Fengjuan Zhu 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.
In order to characterize the bivariate signals, minimum-energy bivariate wavelet frames with arbitrary dilation matrix are studied, which are based on superiority of the minimum-energy frame and the significant properties of bivariate wavelet. Firstly, the concept of minimum-energy bivariate wavelet frame is defined, and its equivalent characterizations and a necessary condition are presented. Secondly, based on polyphase form of symbol functions of scaling function and wavelet function, two sufficient conditions and an explicit constructed method are given. Finally, the decomposition algorithm, reconstruction algorithm, and numerical examples are designed.
Frames theory is one of the efficient tools in the signal processing. It was introduced by Duffin and Schaeffer  and was used to deal with problems in nonharmonic Fourier series. However, people did not pay enough attention to frames theory for a long time. When wavelets theory was booming, Daubechies et al.  defined the affine frame (wavelet frame) by combining the theory of continuous wavelet transform with frame. After that, people started to research frames and its application again. Benedetto and Li  gave the definition of frame multiresolution analysis (FMRA), and their work laid the foundation for other people to do further research. Frames not only can overcome the disadvantages of wavelets and multivariate wavelets but also increase redundancy, and the numerical computation becomes much more stable using frames to reconstruct signal. With well time-frequency localization and shift invariance, frames can be designed more easily than wavelets or multivariate wavelets. At present, frames theory has been widely used in theoretical and applicable domains [4–18], such as signal analysis, image processing, numerical calculation, Banach space theory, and Besov space theory.
In 2000, Chui and He  proposed the concept of minimum-energy wavelet frames. The minimum-energy wavelet frames reduce the computational, and maintain the numerical stability, and do not need to search dual frames in the decomposition and reconstruction of functions (or signals). Therefore, many people paid more attention to the study of minimum-energy wavelet frames. Petukhov  studied the (minimum-energy) wavelet frames with symmetry. Huang and Cheng  studied the construction and characterizations of the minimum-energy wavelet frames with arbitrary integer dilation factor. Gao and Cao  researched the structure of the minimum-energy wavelet frames on the interval [0,1] and its application on signal denoising. Liang and Zhao  studied the minimum-energy multiwavelet frames with dilation factor 2 and multiplicity 2 and gave a characterization and a necessary condition of minimum-energy multiwavelets frames. Huang et al. [10, 11] studied minimum-energy multiwavelet frames and wavelet frames on the interval [0,1] with arbitrary dilation factor. It was well known that a majority of real-world signals are multidimensional, such as graphic and video signal. For this reason, many people studied multivariate wavelets and multivariate wavelet frames [12–18]. In this paper, in order to deal with multidimensional signals and combine organically the minimum-energy wavelet frames with the significant properties of multivariate wavelets, minimum-energy bivariate wavelet frames with arbitrary dilation matrix are studied.
The organization of this paper is as follows. In Section 2, we give preliminaries and basic definitions. Then, in Section 3, the main results are described. In Section 4, we present the decomposition and reconstruction formulas of minimum-energy bivariate wavelet frames. Finally, numerical examples are given in Section 5.
2. Preliminaries and Basic Definitions
2.1. Basic Concept and Notation
Let us recall the concept of dilation matrix and give some notations.
Definition 1 (see ). Let be the integer matrix. Suppose that its eigenvalues have a modulus strictly greater than 1, then is called a dilation matrix.(1)Throughout this paper, let , denote the set of integers and real numbers, respectively. , denote the set of 2-tuple integers and two-dimensional Euclidean space, respectively. For a given dilation matrix , let be a complete set of representatives of , where .(2)Let and . For any , the inner product, norm, and the Fourier transform are defined, respectively, by
(3)For any function , is defined by
(4)We sort the elements of by lexicographical order. That is, for any , and let
then denotes that every component of is zero, denotes that the first nonzero component of is positive, and denotes that the first nonzero component of is negative.(5)For any , are defined, respectively, by
(6)For any , is defined by
where is the -th column of .
Now, we give the definition of frame.
Definition 2 (see ). Let be a complex and separable Hilbert space. A sequence is a frame for if there exist constants , such that, for any ,
The numbers are called frame bounds.
A frame is tight if we can choose as frame bounds in the Definition 2. If a frame ceases to be a frame when an arbitrary element is removed, it is called an exact frame. When , then
2.2. Minimum-Energy Bivariate Wavelet Frame
Definition 3 (see ). A bivariate frame multiresolution analysis (FMRA) for consists of a sequence of closed subspaces and a function such that(1), ;(2);(3);(4) is a frame for ,
where function is called scaling function of FMRA.
Since , satisfies two-scale equation (refinable equation) where . The Fourier transform of (8) is where where is the symbol function of scaling function , .
For simplicity, in this paper, we suppose that any symbol function is trigonometric polynomial, and scaling function and wavelet function are compactly supported.
Definition 4. Let satisfies and continuous at and . Suppose that generates a sequence of nested closed subspaces , then is called a minimum-energy bivariate wavelet frame associated with if for all :
By the Parseval identity, minimum-energy bivariate wavelet frame must be a tight frame for with frame bound being equal to 1. At the same time, the formula (11) is equivalent to The interpretation of minimum-energy bivariate wavelet frame will be clarified later.
Consider , with Using Fourier transform on the previous equation, we can get their symbols as follows: where .
With , we formulate the matrix : and denotes the complex conjugate of the transpose of .
3. Main Result
In this section, we give a complete characterization of minimum-energy bivariate wavelet frames with arbitrary dilation matrix and two sufficient conditions and a necessary condition of minimum-energy bivariate wavelet frame associated with the given scaling function.
Proposition 5. Suppose that is a dilation matrix; let then, is invertible matrix.
The following theorem presents the equivalent characterizations of the minimum-energy bivariate wavelet frames with arbitrary dilation matrix.
Theorem 6. Suppose that the symbols , in (10) and (14) are Laurent polynomial, and generate the refinable function and . If is continuous at and and generates a nested closed subspaces sequence , then the following statements are equivalent.(1) is a minimum-energy bivariate wavelet frame with arbitrary dilation matrix associated with .(2)(3)
Proof. By using the two-scale relations (8) and (13) and notation , then formula (12) is equivalent to
On the other hand, formula (17) can be reformulated as
which is equivalent to
where . Since is an invertible matrix, the previous equation is equivalent to
We multiply the identities in (23) by , respectively, where , and we get let . Take the Fourier transform on the two sides of the previous formula, (23) is equivalent to Thus,
By using the two-scale relations (8) and (13), we can rewrite formula (26) as In other words, the proof of Theorem 6 reduces to the proof of the equivalences of (18), (19), and (27).
It is clear that (18) (27) (19). In order to prove (19) (18), let be any compactly supported function. Let Then by using the properties that, for every fixed , except for finitely many , and both and have compact support, it is clear that only finitely many of the values is nonzero. Now, since is a nontrivial function, by taking the Fourier transform of (19), it follows that the trigonometric polynomial . Obviously, . By choosing , we get formula (27).
By taking the Fourier transform of (27), we get . The proof of Theorem 6 is completed.
Theorem 6 characterizes the necessary and sufficient condition for the existence of the minimum-energy bivariate wavelet frames associated with . But it is not a good choice to use this theorem to construct the minimum-energy bivariate wavelet frames with arbitrary dilation matrix. For convenience, we need to present some sufficient conditions in terms of the symbol functions.
Theorem 7. A compactly supported refinable function , with continuous at and , and is the symbol function of . Let be the minimum-energy multiwavelet frames associated with ; then
Proof. Let be the first column of and . Then,
Since is a Hermitian matrix, the matrix is positive semidefinite.
We have Therefore, that is,
The proof of Theorem 7 is completed.
According to the Theorem 7, there may not exist minimum-energy bivariate wavelet frame associated with a given scaling function. If there exists a minimum-energy bivariate wavelet frame, then the symbol function of scaling function must satisfy (29). Based on the polyphase forms of , we give two sufficient conditions.
Let ; then where . Similarly, where . Let thus, Since is a unitary matrix, condition (17) is equivalent to
For convenience, let , condition (29) can be rewritten as .
If there exists such that then, we have the following Theorem 8.
Theorem 8. Let be a compactly supported refinable function, with continuous at 0 and , and its symbol function satisfies If there exists such that then there exists a minimum-energy bivariate wavelet frame associated with .
Proof. Under the assumption, we know that the vector
is a unit vector.
Construct diagonal matrix such that where with and . It is clear that is a unit vector: and consequently . We next consider the Householder matrix: where , with , and the + or − signs are so chosen that . Then Since Householder matrix is orthogonal matrix, we have By the previous equation, the first component of is . Now , we construct diagonal matrix , such that is also a unit vector and .
Similarly, we define the Householder matrix: where , and such that is also a unit vector and .
Since every component of is a finite sum, we repeat this procedure finite times to get some unitary matrices , That is, is the first column of the unitary matrix Let then, satisfies (38). In the formula (15), we let Then,we can obtain the formula (17). The proof of Theorem 8 is completed.
Corollary 9. Let be a compactly supported refinable function, with continuous at 0 and , and its symbol function satisfies If there exists , such that then there exists a minimum-energy bivariate wavelet frame associated with .
Next, we present an explicit formula of constructing minimum-energy bivariate wavelet frame. Suppose that satisfies (29); let Then are the eigenvalues and eigenvectors of , respectively, where ensure that are unit vector. Let then,
Theorem 10. Let be a compactly supported refinable function, with continuous at 0 and , and its symbol function satisfies If there exists satisfies then the symbol function of minimum-energy bivariate wavelet frame associated with is the first row of the matrix: where is any unitary matrix, and are defined in (59).
Proof. By Theorem 8, there exists a minimum-energy bivariate wavelet frame associated with the given scaling function, and the existence of guarantees that the elements of are trigonometric polynomial, and we have thus, symbol functions satisfy (17).
4. Decomposition and Reconstruction Formulas of Minimum-Energy Bivariate Wavelet Frames
Suppose that the bivariate scaling function has an associated minimum-energy bivariate wavelet frame . Let the projection operators of onto the nested subspace be defined by Then the formula (12) can be rewritten as In other words, the error term between consecutive projections is given by the frame expansion:
Suppose that the error term has another expansion in terms of the frames , that is, Then by using both (67) and (68), we have and this derives This inequality means that the coefficients of the error term in (67) have minimal -norm among all sequences which satisfy (68).
We next discuss minimum-energy bivariate wavelet frames decomposition and reconstruction. For any , we define the coefficients Then we can derive the decomposition and reconstruction formulas that are similar to those of orthonormal wavelets.
5. Numerical Examples
In this section, we present some numerical examples to show the effectiveness of the proposed methods.
Example 1. Let ; then and
Suppose that the Fourier transform of scaling function is Then its symbol function satisfies Thus, Let Then we have . By Corollary 9, there exists a minimum-energy bivariate wavelet frame associated with . Using Theorem 8, let Thus, we have
Consequently, we obtain symbol functions The functions generated by the previous symbol functions are minimum-energy bivariate wavelet frame functions associated with .
Example 2. Let ; then and Suppose that the Fourier transform of scaling function is Then its symbol function satisfies Thus, Let Then we have . By Corollary 9, there exists a minimum-energy bivariate wavelet frame associated with .
Example 3. Let ; then and
Suppose that the Fourier transform of scaling function satisfies Then its symbol function satisfies Let , and we have , . Therefore, Let and we have . By Corollary 9, there exists a minimum-energy bivariate wavelet frame associated with .
In this paper, we define the concept of minimum-energy bivariate wavelet frame with arbitrary dilation matrix and present its equivalent characterizations. We give a necessary condition and two sufficient conditions for minimum-energy bivariate wavelets frame and deduce the decomposition and reconstruction formulas of minimum-energy bivariate wavelets frame. Finally, we give several numerical examples to show the effectiveness of the proposed methods.
The authors would like to thank Professor Weili Li and the anonymous reviewers for suggesting many helpful improvements to this paper. This work was supported by the National Natural Science Foundation of China under the Project Grant no. 61261043, Natural Science Foundation of Ningxia under the Project the Grant no. NX13xxx, and the Research Project of Beifang University of Nationalities under the Project no. 2012Y036.
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