Minimum-Energy Bivariate Wavelet Frame with Arbitrary Dilation Matrix

In order to characterize the bivariate signals, minimum-energy bivariate wavelet frames with arbitrary dilation matrix are studied, which are based on superiority of theminimum-energy frame and the significant properties of bivariate wavelet. Firstly, the concept ofminimum-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.


Introduction
Frames theory is one of the efficient tools in the signal processing.It was introduced by Duffin and Schaeffer [1] 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. [2] 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 [3] 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 timefrequency 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][5][6][7][8][9][10][11][12][13][14][15][16][17][18], such as signal analysis, image processing, numerical calculation, Banach space theory, and Besov space theory.
In 2000, Chui and He [5] 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 [6] studied the (minimum-energy) wavelet frames with symmetry.Huang and Cheng [7] studied the construction and characterizations of the minimum-energy wavelet frames with arbitrary integer dilation factor.Gao and Cao [8] researched the structure of the minimum-energy wavelet frames on the interval [0,1] and its application on signal denoising.Liang and Zhao [9] 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][13][14][15][16][17][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. Definition 1 (see [18]).Let  be the 2 × 2 integer matrix.Suppose that its eigenvalues have a modulus strictly greater than 1, then  is called a dilation matrix.

Preliminaries and Basic Definitions
(1) Throughout this paper, let Z, R denote the set of integers and real numbers, respectively.Z 2 , R 2 denote the set of 2-tuple integers and two-dimensional Euclidean space, respectively.For a given dilation matrix , let {  ,  = 0, 1, . . .,  − 1} be a complete set of representatives of Z 2 /Z 2 , where  = | det()|. ( ), the inner product, norm, and the Fourier transform are defined, respectively, by (3) For any function  ∈  2 (R 2 ),  , () is defined by (4) We sort the elements of Z 2 by lexicographical order.That is, for any  = ( 1 ,  2 )  ,  = ( 1 ,  2 )  , ,  ∈ Z 2 , 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.
By the Parseval identity, minimum-energy bivariate wavelet frame Ψ must be a tight frame for  2 (R 2 ) 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.

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 𝐴 is a dilation matrix; let
then, A is invertible matrix.
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  ∈  2 (R 2 ), with φ continuous at 0 and φ(0) = 1, and  0 () is the symbol function of .Let Ψ = { 1 ,  2 , . . .,   } be the minimum-energy multiwavelet frames associated with ; then Proof.Let () be the first column of () and () = ((), ()).Then,  ()  * () +  ()  * () =   .
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  0 (),  1 (), . . .,   (), we give two sufficient conditions.
Theorem 8. Let () ∈  2 (R 2 ) be a compactly supported refinable function, with φ continuous at 0 and φ(0) = 1, and its symbol function satisfies If there exists  0  () 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  0 = diag(  0 ,   1 , . . .,    ) such that where a  ∈ R +1 with a 0 ̸ = 0 and a  ̸ = 0.It is clear that f 1 is a unit vector: and consequently a  0 a  = 0. We next consider the ( + 1) × ( + 1) Householder matrix: where k = a  ± ‖a  ‖e 1 , with e 1 = (1, 0, . . ., 0)  +1 , and the + or − signs are so chosen that k ̸ = 0. Then Since Householder matrix is orthogonal matrix, we have By the previous equation, the first component of  1 a 0 is 0.
Proof.By Theorem 8, there exists a minimum-energy bivariate wavelet frame associated with the given scaling function, and the existence of  0  () guarantees that the elements of √Λ() are trigonometric polynomial, and we have

Decomposition and Reconstruction Formulas of Minimum-Energy Bivariate Wavelet Frames
Suppose that the bivariate scaling function () has an associated minimum-energy bivariate wavelet frame { 1 , . . .,   }.
Let the projection operators P  of  2 (R 2 ) onto the nested subspace   be defined by Then the formula (12) can be rewritten as In other words, the error term   = P +1  − P   between consecutive projections is given by the frame expansion: Suppose that the error term   has another expansion in terms of the frames { 1 , . . .,   }, that is, Then by using both (67) and (68), we have

Numerical Examples
In this section, we present some numerical examples to show the effectiveness of the proposed methods.

2. 1 .
Basic Concept and Notation.Let us recall the concept of dilation matrix and give some notations.