Minimum-Energy Bivariate Wavelet Frame with Arbitrary Dilation Matrix

Abstract

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.

1. 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 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 [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 [12–18]. 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 [18]). Let 𝐴 be the 2×2 integer matrix. Suppose that its eigenvalues have a modulus strictly greater than 1, then 𝐴 is called a dilation matrix.(1)Throughout this paper, let ℤ, ℝ denote the set of integers and real numbers, respectively. ℤ², ℝ² denote the set of 2-tuple integers and two-dimensional Euclidean space, respectively. For a given dilation matrix 𝐴, let {𝜇_𝑖,𝑖=0,1,…,𝑠−1} be a complete set of representatives of ℤ²/𝐴ℤ², where 𝑠=|det(𝐴)|.(2)Let 𝐿²(ℝ²)={𝑓∶∫_ℝ2|𝑓(𝑥)|²𝑑𝑥<∞} and 𝑙²(ℤ²)={𝑠∶∑_𝑚∈ℤ²|𝑠_𝑚|²𝑑𝑥<∞}. For any 𝑓,𝑔∈𝐿²(ℝ²), the inner product, norm, and the Fourier transform are defined, respectively, by 𝑓(𝑥),𝑔(𝑥)=_ℝ2𝑓(𝑥)𝑔(𝑥)𝑑𝑥,𝑓²=,𝑓(𝜉)=1𝑓,𝑓√2𝜋^ℝ2𝑓(𝑥)𝑒^{−𝑖𝑥⋅𝜉}𝑑𝑥. (1)(3)For any function 𝑓∈𝐿²(ℝ²), 𝑓_𝑗,𝑘(𝑥) is defined by 𝑓_𝑗,𝑘(𝑥)=𝑠^𝑗/2𝑓𝐴^𝑗𝑥−𝑘,𝑗∈ℤ,𝑘∈ℤ². (2)(4)We sort the elements of ℤ² by lexicographical order. That is, for any 𝑘=(𝑘₁,𝑘₂)^𝑇,𝑙=(𝑙₁,𝑙₂)^𝑇,𝑘,𝑙∈ℤ², and let 𝑚=𝑘₁−𝑙₁,𝑘₂−𝑙₂; (3)then 𝑘=𝑙 denotes that every component of 𝑚 is zero, 𝑘>𝑙 denotes that the first nonzero component of 𝑚 is positive, and 𝑘<𝑙 denotes that the first nonzero component of 𝑚 is negative.(5)For any 𝑧=(𝑧₁,𝑧₂)^𝑇,𝑧₁≠0,𝑧₂≠0,𝑘=(𝑘₁,𝑘₂)^𝑇∈ℝ², 𝑧⁻¹,𝑧^𝑘 are defined, respectively, by 𝑧⁻¹=𝑧₁⁻¹,𝑧₂^−1𝑇,𝑧^𝑘=²_𝑖=1𝑧^𝑘𝑖_𝑖. (4)(6)For any 𝐴∈ℝ^2×2, 𝑧^𝐴 is defined by 𝑧^𝐴=𝑧^𝐴1,𝑧^𝐴2𝑇, (5)where 𝐴_𝑖 is the 𝑖-th column of 𝐴.
Now, we give the definition of frame.

Definition 2 (see [18]). Let ℍ be a complex and separable Hilbert space. A sequence {𝑓_𝑘∶𝑘∈ℤ} is a frame for ℍ if there exist constants 0<𝐴,𝐵<∞, such that, for any 𝑓∈ℍ, 𝐴𝑓²≤_𝑘∈ℤ𝑓,𝑓_𝑘²≤𝐵𝑓². (6) The numbers 𝐴,𝐵 are called frame bounds.
A frame is tight if we can choose 𝐴=𝐵 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 𝐴=𝐵=1, then 𝑓=_𝑘∈ℤ𝑓,𝑓_𝑘𝑓_𝑘,∀𝑓∈ℍ. (7)

2.2. Minimum-Energy Bivariate Wavelet Frame

Definition 3 (see [18]). A bivariate frame multiresolution analysis (FMRA) for 𝐿²(ℝ²) consists of a sequence of closed subspaces {𝑉_𝑗}_𝑗∈ℤ and a function 𝜙∈𝑉₀ such that(1)𝑉_𝑗⊂𝑉_𝑗+1,  𝑗∈ℤ;(2)⋂_𝑗∈ℤ𝑉_𝑗={0},⋃_𝑗∈ℤ𝑉_𝑗=𝐿²(ℝ²);(3)𝑓(𝑥)∈𝑉_𝑗⇔𝑓(𝐴𝑥)∈𝑉_𝑗+1,𝑗∈ℤ,𝑥∈ℝ²;(4){𝜙(𝑥−𝑛)}_𝑛∈ℤ² is a frame for 𝑉₀,
where function 𝜙(𝑥) is called scaling function of FMRA.
Since 𝑉₀⊂𝑉₁, 𝜙(𝑥) satisfies two-scale equation (refinable equation) 𝜙(𝑥)=_𝑘∈ℤ2𝑝_𝑘𝜙(𝐴𝑥−𝑘), (8) where {𝑝_𝑘}_𝑘∈ℤ²∈𝑙²(ℤ²). The Fourier transform of (8) is 𝜙(𝜉)=𝑚₀𝐴^−𝑇𝜉𝜙𝐴^−𝑇𝜉,𝜉∈ℝ², (9) where 𝑚₀(𝜉)=1|det(𝐴)|_𝑘∈ℤ2𝑝_𝑘𝑒^{−𝑖𝜉𝑇𝑘}=1𝑠_𝑘∈ℤ2𝑝_𝑘𝑧^𝑘, (10) where 𝑚₀(𝜉) is the symbol function of scaling function 𝜙(𝑥), 𝑧=𝑒^{−𝑖𝜉𝑇}.
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 𝜙∈𝐿²(ℝ²) satisfies 𝜙∈𝐿^∞ and 𝜙 continuous at 0 and 𝜙(0)=1. Suppose that 𝜙 generates a sequence of nested closed subspaces {𝑉_𝑗}_𝑗∈ℤ, then Ψ={𝜓¹,𝜓²,…,𝜓^𝑁}⊂𝑉₁ is called a minimum-energy bivariate wavelet frame associated with 𝜙 if for all 𝑓∈𝐿²(ℝ²): _𝑘∈ℤ2𝑓,𝜙_1,𝑘²=_𝑘∈ℤ2𝑓,𝜙_0,𝑘²+^𝑁_{𝑙=1𝑘∈ℤ2}𝑓,𝜓^𝑙_0,𝑘². (11)
By the Parseval identity, minimum-energy bivariate wavelet frame Ψ must be a tight frame for 𝐿²(ℝ²) with frame bound being equal to 1. At the same time, the formula (11) is equivalent to _𝑘∈ℤ2𝑓,𝜙_1,𝑘𝜙_1,𝑘=_𝑘∈ℤ2𝑓,𝜙_0,𝑘𝜙_0,𝑘+^𝑁_{𝑙=1𝑘∈ℤ2}𝑓,𝜓^𝑙_0,𝑘𝜓^𝑙_0,𝑘,∀𝑓∈𝐿²ℝ². (12) The interpretation of minimum-energy bivariate wavelet frame will be clarified later.
Consider Ψ={𝜓¹,𝜓²,…,𝜓^𝑁}⊂𝑉₁, with 𝜓^𝑙(𝑥)=_𝑘∈ℤ2𝑞^𝑙_𝑘𝜙(𝐴𝑥−𝑘),𝑙=1,2,…,𝑁. (13) Using Fourier transform on the previous equation, we can get their symbols as follows: 𝑚_𝑙(𝜉)=1𝑠_𝑘∈ℤ2𝑞^𝑙_𝑘𝑒^{−𝑖𝜉𝑇𝑘}=1𝑠_𝑘∈ℤ2𝑞^𝑙_𝑘𝑧^𝑘,𝑙=1,2,…,𝑁, (14) where 𝑧=𝑒^{−𝑖𝜉𝑇}.
With 𝑚₀(𝜉),𝑚₁(𝜉),…,𝑚_𝑁(𝜉), we formulate the 𝑠×(𝑁+1) matrix 𝑀(𝜉): 𝑀(𝜉)=𝑚₀(𝜉)𝑚₁(𝜉)⋯𝑚_𝑁(𝜉)𝑚₀𝜉+2𝐴^−𝑇𝜇₁𝜋𝑚₁𝜉+2𝐴^−𝑇𝜇₁𝜋⋯𝑚_𝑁𝜉+2𝐴^−𝑇𝜇₁𝜋𝑚⋮⋮⋮₀𝜉+2𝐴^−𝑇𝜇_𝑠−1𝜋𝑚₁𝜉+2𝐴^−𝑇𝜇_𝑠−1𝜋⋯𝑚_𝑁𝜉+2𝐴^−𝑇𝜇_𝑠−1𝜋, (15) and 𝑀^∗(𝜉) denotes the complex conjugate of the transpose of 𝑀(𝜉).

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 𝐴 is a dilation matrix; let 𝑒𝒜=^{𝑖2𝜋𝜇𝑇1𝐴−1𝜇1}𝑒^{𝑖2𝜋𝜇𝑇1𝐴−1𝜇2}⋯𝑒^{𝑖2𝜋𝜇𝑇1𝐴−1𝜇𝑠−1}𝑒^{𝑖2𝜋𝜇𝑇2𝐴−1𝜇1}𝑒^{𝑖2𝜋𝜇𝑇2𝐴−1𝜇2}⋯𝑒^{𝑖2𝜋𝜇𝑇2𝐴−1𝜇𝑠−1}𝑒⋮⋮⋮_{𝑖2𝜋𝜇𝑇𝑠−1𝐴−1𝜇1}𝑒^{𝑖2𝜋𝜇𝑇𝑠−1𝐴−1𝜇2}⋯𝑒^{𝑖2𝜋𝜇𝑇𝑠−1𝐴−1𝜇𝑠−1}; (16) then, 𝒜 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 𝑚_𝑙(𝜉), 𝑙=0,1,…,𝑁 in (10) and (14) are Laurent polynomial, and generate the refinable function 𝜙(𝑥) and Ψ={𝜓¹,𝜓²,…,𝜓^𝑁}. If 𝜙 is continuous at 0 and 𝜙(0)=1 and 𝜙(𝑥) generates a nested closed subspaces sequence {𝑉_𝑗}_𝑗∈𝐙, then the following statements are equivalent.(1)Ψ is a minimum-energy bivariate wavelet frame with arbitrary dilation matrix 𝐴 associated with 𝜙(𝑥).(2)𝑀(𝜉)𝑀^∗(𝜉)=𝐼_𝑠,∀𝜉∈ℝ². (17)(3)𝛼_𝑚𝑙=_𝑘∈ℤ2𝑝^∗_{𝑚−𝐴𝑘}𝑝_{𝑙−𝐴𝑘}+^𝑁_𝑖=1𝑞^𝑖∗_{𝑚−𝐴𝑘}𝑞^𝑖_{𝑙−𝐴𝑘}−𝑠𝛿_𝑚𝑙=0,∀𝑚,𝑙∈ℤ². (18)

Proof. By using the two-scale relations (8) and (13) and notation 𝛼_𝑚𝑙, then formula (12) is equivalent to _{𝑚,𝑙∈ℤ2}𝛼_𝑚𝑙𝑓,𝜙(𝐴𝑥−𝑚)𝜙(𝐴𝑥−𝑙)=0,∀𝑓∈𝐿²ℝ². (19) On the other hand, formula (17) can be reformulated as ^𝑁_𝑙=0𝑚_𝑙(𝜉)²=1,_𝑁𝑙=0𝑚_𝑙(𝜉)𝑚^∗_𝑙𝜉+2𝐴⁻¹𝜇_𝑗𝜋=0,𝑗=1,…,𝑠−1 (20) which is equivalent to ^𝑁_𝑙=0𝑚_𝑙(𝜉)^𝑠−1_𝑗=0𝑚^∗_𝑙𝜉+2𝐴⁻¹𝜇_𝑗𝜋=1,_𝑁𝑙=0𝑚_𝑙(𝜉)𝑚^∗₀(𝜉)−^𝑠−1_𝑗=1𝑚^∗_𝑙𝜉+2𝐴⁻¹𝜇_𝑗𝜋=1,_𝑁𝑙=0𝑚_𝑙(𝜉)^𝑠−1_𝑗=0𝑚^∗_𝑙𝜉+2𝐴⁻¹𝜇_𝑗𝜋−2𝑚_𝑙𝜉+2𝐴⁻¹𝜇_𝑛𝜋=1,𝑛=1,…,𝑠−1;𝜉∈ℝ², (21) or ^𝑁_𝑙=0𝑚_𝑙(𝜉)_𝑘∈ℤ2𝑞^𝑙∗_−𝐴𝑘𝑧^𝐴𝑘=1,_𝑁𝑙=0𝑚_𝑙(𝜉)^𝑠−1_{𝑛=1𝑘∈ℤ2}𝑞_𝜇^𝑙∗_{𝑛−𝐴𝑘}𝑧^{𝐴𝑘−𝜇𝑛}=𝑠−1,_𝑁𝑙=0𝑚_𝑙(𝜉)^𝑠−1_𝑛=1𝑒^{𝑖2𝜇𝑇𝑗𝐴−1𝜇𝑛𝜋}_𝑘∈ℤ2𝑞_𝜇^𝑙∗_{𝑛−𝐴𝑘}𝑧^{𝐴𝑘−𝜇𝑛}=−1,𝑗=1,…,𝑠−1;𝜉∈ℝ², (22) where 𝑞⁰_𝑘=𝑝_𝑘,𝑘∈ℤ². Since 𝒜 is an invertible matrix, the previous equation is equivalent to ^𝑁_𝑙=0𝑚_𝑙(𝜉)_𝑘∈ℤ2𝑞_𝜇^𝑙∗_{𝑛−𝐴𝑘}𝑧^{𝐴𝑘−𝜇𝑛}=1,𝑛=0,1,…,𝑠−1. (23)
We multiply the identities in (23) by 𝑧^𝜇𝑛𝜙(𝐴^−𝑇𝜉),𝑛=0,1,…,𝑠−1, respectively, where 𝑧=𝑒^{−𝑖𝜉𝑇𝐴−1}, and we get 𝜙𝐴^−𝑇𝜉𝑧^𝜇𝑛=^𝑁_{𝑙=0𝑘∈ℤ2}𝑞_𝜇^𝑙∗_{𝑛−𝐴𝑘}𝑧^𝐴𝑘𝜓^𝑙(𝜉),𝑛=0,1,…,𝑠−1; (24) let 𝜓⁰=𝜙. Take the Fourier transform on the two sides of the previous formula, (23) is equivalent to 𝑠𝜙𝐴𝑥−𝜇_𝑛=^𝑁_{𝑙=0𝑘∈ℤ2}𝑞_𝜇^𝑙∗_{𝑛−𝐴𝑘}𝜓^𝑙(𝑥−𝑘),𝑛=0,1,…,𝑠−1. (25) Thus, 𝑠𝜙𝐴𝑥−𝜇=^𝑁_{𝑙=0𝑘∈ℤ2}𝑞^𝑙∗_{𝜇−𝐴𝑘}𝜓^𝑙(𝑥−𝑘),∀𝜇∈ℤ². (26)
By using the two-scale relations (8) and (13), we can rewrite formula (26) as _𝑚∈ℤ2𝛼_𝑚𝑙𝜙(𝐴𝑥−𝑚)=0,∀𝑙∈ℤ². (27) 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 𝑓∈𝐿²(ℝ²) be any compactly supported function. Let 𝛽_𝑙𝑓=𝑓,_𝑚∈ℤ2𝛼_𝑚𝑙𝜙(𝐴𝑥−𝑚),𝑙∈ℤ². (28) Then by using the properties that, for every fixed 𝑚, 𝛼_𝑚𝑙=0 except for finitely many 𝑙, and both 𝜙 and 𝑓 have compact support, it is clear that only finitely many of the values 𝛽_𝑙(𝑓) is nonzero. Now, since 𝜙(𝜉) is a nontrivial function, by taking the Fourier transform of (19), it follows that the trigonometric polynomial ∑_𝑙∈ℤ²𝛽_𝑙(𝑓)𝑒^{−𝑖𝜉𝑇𝐴−𝑇𝑙}≡0. Obviously, 𝛽_𝑙(𝑓)=0,𝑙∈ℤ². By choosing 𝑓=∑_𝑚∈ℤ²𝛼_𝑚𝑙𝜙(𝐴𝑥−𝑚), we get formula (27).
By taking the Fourier transform of (27), we get 𝛼_𝑚𝑙=0,𝑚,𝑙∈ℤ². 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 𝜙. 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 𝜙∈𝐿²(ℝ²), with 𝜙 continuous at 0 and 𝜙(0)=1, and 𝑚₀(𝜉) is the symbol function of 𝜙. Let Ψ={𝜓¹,𝜓²,…,𝜓^𝑁} be the minimum-energy multiwavelet frames associated with 𝜙; then ^𝑠−1_𝑛=0𝑚₀𝜉+2𝐴⁻¹𝜇_𝑛𝜋²≤1,∀𝜉∈ℝ². (29)

Proof. Let 𝐷(𝜉) be the first column of 𝑀(𝜉) and 𝑀(𝜉)=(𝐷(𝜉),𝑄(𝜉)). Then, 𝐷(𝜉)𝐷^∗(𝜉)+𝑄(𝜉)𝑄^∗(𝜉)=𝐼_𝑠. (30) Since 𝑄(𝜉)𝑄^∗(𝜉) is a Hermitian matrix, the matrix 𝐼_𝑠−𝐷(𝜉)𝐷^∗(𝜉) is positive semidefinite.
We have 𝐼_𝑠𝐷(𝜉)𝐷^∗(𝜉)1𝐼_𝑠−𝐷(𝜉)−𝐷^∗(𝜉)1=𝐼_𝑠−𝐷(𝜉)𝐷^∗(𝜉)1−𝐷^∗(𝜉)𝐷(𝜉)𝐼det^𝑠𝐷(𝜉)𝐷^∗(𝜉)1𝐼=det^𝑠𝐷(𝜉)1−𝐷^∗(𝜉)𝐷(𝜉),𝐼det^𝑠−𝐷(𝜉)−𝐷^∗(𝜉)1𝐼=det^𝑠−𝐷(𝜉)1−𝐷^∗(𝜉)𝐷(𝜉). (31) Therefore, det𝐼_𝑠−𝐷(𝜉)𝐷^∗(𝜉)1−𝐷^∗(𝜉)𝐷(𝜉)=1−𝐷^∗(𝜉)𝐷(𝜉)1−𝐷^∗(𝜉)𝐷(𝜉), (32) that is, 1−𝐷^∗(𝜉)𝐷(𝜉)≥0. (33)
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 𝑚₀(𝑧),𝑚₁(𝑧),…,𝑚_𝑁(𝑧), we give two sufficient conditions.

Let 𝑧=𝑒^{−𝑖𝜉𝑇}; then 𝑚₀(𝜉)=1𝑠_𝑘∈ℤ2𝑝_𝑘𝑒^{−𝑖𝜉𝑇𝑘}=1𝑠^𝑠−1_{𝑖=0𝑘∈ℤ2}𝑝_{𝜇𝑖+𝐴𝑘}𝑒^{−𝑖𝜉𝑇(𝜇𝑖+𝐴𝑘)}=1𝑠^𝑠−1_𝑖=0𝑒^{−𝑖𝜉𝑇𝜇𝑖}_𝑘∈ℤ2𝑝_{𝜇𝑖+𝐴𝑘}𝑒^{−𝑖𝜉𝑇𝐴𝑘}=1𝑠^𝑠−1_𝑖=0𝑧^𝜇𝑖_𝑘∈ℤ2𝑝_{𝜇𝑖+𝐴𝑘}𝑧^𝐴𝑘=1√𝑠^𝑠−1_𝑖=0𝑧^𝜇𝑖𝑓⁰_𝑖𝑧^𝐴, (34) where 𝑓⁰_𝑖(𝑥)=(1/√𝑠)∑_𝑘∈ℤ²𝑝_{𝜇𝑖+𝐴𝑘}𝑥^𝑘,𝑥∈ℝ². Similarly, 𝑚_𝑙(𝜉)=1√𝑠^𝑠−1_𝑖=0𝑧^𝜇𝑖𝑓^𝑙_𝑖𝑧^𝐴,𝑙=1,2,…,𝑁, (35) where 𝑓^𝑙_𝑖(𝑥)=(1/√𝑠)∑_𝑘∈ℤ²𝑞^𝑙_{𝜇𝑖+𝐴𝑘}𝑥^𝑘,𝑥∈ℝ². Let 𝐶(𝑧)=1√𝑠×𝑧^𝜇0𝑒^{−𝑖2𝜋𝜇𝑇0𝐴−1𝜇0}𝑧^𝜇1𝑒^{−𝑖2𝜋𝜇𝑇0𝐴−1𝜇1}⋯𝑧^𝜇𝑠−1𝑒^{−𝑖2𝜋𝜇𝑇0𝐴−1𝜇𝑠−1}𝑧^𝜇0𝑒^{−𝑖2𝜋𝜇𝑇1𝐴−1𝜇0}𝑧^𝜇1𝑒^{−𝑖2𝜋𝜇𝑇1𝐴−1𝜇1}⋯𝑧^𝜇𝑠−1𝑒^{−𝑖2𝜋𝜇𝑇1𝐴−1𝜇𝑠−1}𝑧⋮⋮⋮_𝜇0𝑒^{−𝑖2𝜋𝜇𝑇𝑠−1𝐴−1𝜇0}𝑧^𝜇1𝑒^{−𝑖2𝜋𝜇𝑇𝑠−1𝐴−1𝜇1}⋯𝑧^𝜇3𝑒^{−𝑖2𝜋𝜇𝑇𝑠−1𝐴−1𝜇𝑠−1}; (36) thus, 𝑀(𝜉)(𝑧)𝑓=𝐶⁰⁰𝑧^𝐴𝑓¹₀𝑧^𝐴⋯𝑓^𝑁₀𝑧^𝐴𝑓⁰₁𝑧^𝐴𝑓¹₁𝑧^𝐴⋯𝑓^𝑁₁𝑧^𝐴𝑓⋮⋮⋮_0𝑠−1𝑧^𝐴𝑓¹_𝑠−1𝑧^𝐴⋯𝑓^𝑁_𝑠−1𝑧^𝐴. (37) Since 𝐶(𝑧) is a unitary matrix, condition (17) is equivalent to 𝑓⁰₀𝑧^𝐴𝑓¹₀𝑧^𝐴⋯𝑓^𝑁₀𝑧^𝐴𝑓⁰₁𝑧^𝐴𝑓¹₁𝑧^𝐴⋯𝑓^𝑁₁𝑧^𝐴𝑓⋮⋮⋮_0𝑠−1𝑧^𝐴𝑓¹_𝑠−1𝑧^𝐴⋯𝑓^𝑁_𝑠−1𝑧^𝐴×𝑓⁰₀𝑧^𝐴𝑓¹₀𝑧^𝐴⋯𝑓^𝑁₀𝑧^𝐴𝑓⁰₁𝑧^𝐴𝑓¹₁𝑧^𝐴⋯𝑓^𝑁₁𝑧^𝐴𝑓⋮⋮⋮_0𝑠−1𝑧^𝐴𝑓¹_𝑠−1𝑧^𝐴⋯𝑓^𝑁_𝑠−1𝑧^𝐴∗=𝐼_𝑠. (38)

For convenience, let 𝑢=𝑧^𝐴, condition (29) can be rewritten as ∑^𝑠−1_𝑛=0|𝑓⁰_𝑛(𝑢)|²≤1.

If there exists 𝑓⁰_𝑠(𝑢) such that ^𝑠_𝑛=0𝑓⁰_𝑛(𝑢)²=1, (39) then, we have the following Theorem 8.

Theorem 8. Let 𝜙(𝑥)∈𝐿²(ℝ²) be a compactly supported refinable function, with 𝜙 continuous at 0 and 𝜙(0)=1, and its symbol function satisfies ^𝑠−1_𝑛=0𝑓⁰_𝑛(𝑢)²≤1. (40) If there exists 𝑓⁰_𝑠(𝑢) such that ^𝑠_𝑛=0𝑓⁰_𝑛(𝑢)²=1, (41) then there exists a minimum-energy bivariate wavelet frame associated with 𝜙(𝑥).

Proof. Under the assumption, we know that the vector 𝐟∶=𝑓⁰₀(𝑢),𝑓⁰₁(𝑢),…,𝑓⁰_𝑠−1(𝑢),𝑓⁰_𝑠(𝑢)^𝑇 (42) is a unit vector.
Construct diagonal matrix 𝐷₀=diag(𝑢^𝑡0,𝑢^𝑡1,…,𝑢^𝑡𝑠) such that 𝐟₁=𝐷₀𝐟=𝑢^𝑡0𝑓⁰₀(𝑢),𝑢^𝑡1𝑓⁰₁(𝑢),…,𝑢^𝑡𝑠𝑓⁰_𝑠(𝑢)^𝑇=^𝐾_()𝑗=0,0𝐚_𝑗𝑢^𝑗,𝑡₀,𝑡₁,…,𝑡_𝑠∈ℤ², (43) where 𝐚_𝑗∈ℝ^𝑠+1 with 𝐚₀≠𝟎 and 𝐚_𝐾≠𝟎. It is clear that 𝐟₁ is a unit vector: 𝐟^∗₁𝐟₁=^𝐾_𝑗=(0,0)𝐚_𝑗𝑢^𝑗∗𝐾_𝑗=(0,0)𝐚_𝑗𝑢^𝑗=1,∀|𝑢|=1, (44) and consequently 𝐚^𝑇₀𝐚_𝐾=0. We next consider the (𝑠+1)×(𝑠+1) Householder matrix: 𝐻₁=𝐼_𝑠+1−2‖𝐯‖²𝐯𝐯^𝑇, (45) where 𝐯=𝐚_𝐾±‖𝐚_𝐾‖𝐞₁, with 𝐞₁=(1,0,…,0)^𝑇_𝑠+1, and the + or − signs are so chosen that 𝐯≠0. Then 𝐻₁𝐚_𝐾=±𝐚_𝐾𝐞₁. (46) Since Householder matrix is orthogonal matrix, we have 𝐻₁𝐚₀^𝑇𝐻₁𝐚_𝐾=𝐚^𝑇₀𝐻^𝑇₁𝐻₁𝐚_𝐾=𝐚^𝑇₀𝐚_𝐾=0. (47) By the previous equation, the first component of 𝐻₁𝐚₀ is 0. Now 𝐻₁𝐟₁=∑^𝐾_𝑗=(0,0)(𝐻₁𝐚_𝑗)𝑢^𝑗, we construct diagonal matrix 𝐷₁=diag(𝑢^𝑡(1),1,…,1), 𝑡₍₁₎∈ℤ² such that 𝐟₂=𝐷₁𝐻₁𝐟₁=𝐷₁^𝐾_()𝑗=0,0𝐻₁𝐚_𝑗𝑢^𝑗=^𝐾_()𝑗=0,0𝐚_𝑗⁽¹⁾𝑢^𝑗 (48) is also a unit vector and 𝐾₁<𝐾,𝐚₀⁽¹⁾≠𝟎,𝐚_𝐾⁽¹⁾₁≠𝟎.
Similarly, we define the Householder matrix: 𝐻₂=𝐼_𝑠+1−2‖𝐯‖²𝐯𝐯^𝑇, (49) where 𝐯=𝐚_𝐾1±‖𝐚_𝐾1‖𝐞₁≠𝟎, and 𝐷₂=diag(𝑢^𝑡(2),1,…,1) such that 𝐟₃=𝐷₂𝐻₂𝐟₂=𝐷₂^𝐾1_()𝑗=0,0𝐻₂𝐚_𝑗⁽¹⁾𝑢^𝑗=^𝐾1_𝑗=(0,0)𝐚_𝑗⁽²⁾𝑢^𝑗 (50) is also a unit vector and 𝐾₂<𝐾₁,𝐚₀⁽²⁾≠𝟎,𝐚_𝐾⁽²⁾₂≠𝟎.
Since every component of 𝐟 is a finite sum, we repeat this procedure finite times to get some unitary matrices 𝐷_𝐿,𝐻_𝐿,𝐷_𝐿−1,𝐻_𝐿−1,…,𝐻₂,𝐷₁,𝐻₁, 𝐷_𝐿𝐻_𝐿𝐷_𝐿−1𝐻_𝐿−1⋯𝐻₂𝐷₁𝐻₁𝐷₀𝐟=𝐞₁. (51) That is, 𝐟 is the first column of the unitary matrix 𝐻∶=𝐷^∗₀𝐻^∗₀𝐷^∗₁𝐻^∗₂⋯𝐻^∗_𝐿−1𝐷^∗_𝐿−1𝐻^∗_𝐿𝐷^∗_𝐿. (52) Let 𝑓𝐻=⁰⁰(𝑢)𝑓¹₀(𝑢)⋯𝑓^𝑠₀(𝑢)𝑓⁰₁(𝑢)𝑓¹₁(𝑢)⋯𝑓^𝑠₁(𝑢)𝑓⋮⋮⋮_0𝑠(𝑢)𝑓¹_𝑠(𝑢)⋯𝑓^𝑠_𝑠(𝑢); (53) then, 𝐻 satisfies (38). In the formula (15), we let 𝑚_𝑙(𝜉)=1√𝑠^𝑠−1_𝑖=0𝑧^𝜇𝑖𝑓^𝑙_𝑖𝑧^𝐴,𝑙=1,…,𝑠. (54) Then,we can obtain the formula (17). The proof of Theorem 8 is completed.

Corollary 9. Let 𝜙(𝑥)∈𝐿²(ℝ²) be a compactly supported refinable function, with 𝜙 continuous at 0 and 𝜙(0)=1, and its symbol function satisfies ^𝑠−1_𝑛=0𝑚₀𝜉+2𝐴⁻¹𝜇_𝑛𝜋²≤1,∀𝜉∈ℝ². (55) If there exists 𝑓⁰_𝑠(𝑢),…,𝑓⁰_𝑛(𝑢),𝑛≥𝑠, such that ^𝑛_𝑖=0𝑓⁰_𝑖(𝑢)²=1, (56) then there exists a minimum-energy bivariate wavelet frame associated with 𝜙(𝑥).

Next, we present an explicit formula of constructing minimum-energy bivariate wavelet frame. Suppose that 𝑚₀ satisfies (29); let 𝒟(𝜉)=𝐼_𝑠−𝐷(𝜉)𝐷^∗(𝜉). (57) Then 𝜆₁=⋯=𝜆_𝑠−1=1,𝜆_𝑠=1−^𝑠−1_𝑛=0𝑚₀𝜉+2𝐴^−𝑇𝜇_𝑛𝜋².𝛽₁=1Ω₁𝑚^∗₀(𝜉+2𝐴^−𝑇𝜇₁𝜋),−𝑚^∗₀(𝜉+2𝐴^−𝑇𝜇₀𝜋),0,…,0^𝑇𝛽_𝑘=1Ω_𝑘−𝑚₀𝜉+2𝐴^−𝑇𝜇₀𝜋𝑚^∗₀𝜉+2𝐴^−𝑇𝜇_𝑘𝜋,−𝑚₀𝜉+2𝐴^−𝑇𝜇₁𝜋𝑚^∗₀𝜉+2𝐴^−𝑇𝜇_𝑘𝜋,…,−𝑚₀𝜉+2𝐴^−𝑇𝜇_𝑘−1𝜋𝑚^∗₀𝜉+2𝐴^−𝑇𝜇_𝑘𝜋,_{𝑘−1𝑗=0}𝑚₀𝜉+2𝐴^−𝑇𝜇_𝑗𝜋²,0,…,0^𝑇𝛽𝑘=2,3,…,𝑠−1,_𝑠=1Ω_𝑠𝑚₀𝜉+2𝐴^−𝑇𝜇₀𝜋,𝑚₀𝜉+2𝐴^−𝑇𝜇₁𝜋,…,𝑚₀𝜉+2𝐴^−𝑇𝜇_𝑠−1𝜋^𝑇 (58) are the eigenvalues and eigenvectors of 𝒟(𝜉), respectively, where Ω₁,Ω₂,…,Ω_𝑠 ensure that 𝛽₁,𝛽₂,…,𝛽_𝑠 are unit vector. Let 𝑆(𝜉)=𝛽₁𝛽₂⋯𝛽_𝑠,Λ(𝜉)=diag1,…,1,𝜆_𝑠, (59) then, 𝐼_𝑠−𝐷(𝜉)𝐷^∗(𝜉)=𝑆(𝜉)Λ(𝜉)𝑆^∗(𝜉). (60)

Theorem 10. Let 𝜙(𝑥)∈𝐿²(ℝ²) be a compactly supported refinable function, with 𝜙 continuous at 0 and 𝜙(0)=1, and its symbol function satisfies ^𝑠−1_𝑛=0𝑚₀𝜉+2𝐴⁻¹𝜇_𝑛𝜋²≤1,∀𝜉∈ℝ². (61) If there exists 𝑓⁰_𝑠(𝑢) satisfies ^𝑠_𝑖=0𝑓⁰_𝑖(𝑢)²=1, (62) then the symbol function of minimum-energy bivariate wavelet frame associated with 𝜙(𝑥) is the first row of the matrix: 𝑄(𝜉)=𝑈(𝜉)Λ(𝜉)𝑆(𝜉), (63) where 𝑈(𝜉) is any 𝑠×𝑠 unitary matrix, and 𝑆(𝜉),Λ(𝜉) 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 𝑓⁰_𝑠(𝑢) guarantees that the elements of Λ(𝜉) are trigonometric polynomial, and we have 𝑄(𝜉)𝑄^∗(𝜉)=𝑆(𝜉)Λ(𝜉)𝑈(𝜉)𝑆(𝜉)Λ(𝜉)𝑈(𝜉)^∗=𝑆(𝜉)Λ(𝜉)𝑆^∗(𝜉)=𝐼_𝑠−𝐷(𝜉)𝐷^∗(𝜉); (64) thus, symbol functions satisfy (17).

Theorems 8 and 10 give different methods to construct minimum-energy bivariate wavelet frame.

4. Decomposition and Reconstruction Formulas of Minimum-Energy Bivariate Wavelet Frames

Suppose that the bivariate scaling function 𝜙(𝑥) has an associated minimum-energy bivariate wavelet frame {𝜓¹,…,𝜓^𝑁}. Let the projection operators 𝐏_𝑗 of 𝐿²(ℝ²) onto the nested subspace 𝑉_𝑗 be defined by 𝐏_𝑗𝑓∶=_𝑘∈ℤ2𝑓,𝜙_𝑗,𝑘𝜙_𝑗,𝑘. (65) Then the formula (12) can be rewritten as 𝐏_𝑗+1𝑓−𝐏_𝑗𝑓∶=^𝑁_{𝑖=1𝑘∈ℤ2}𝑓,𝜓^𝑖_𝑗,𝑘𝜓^𝑖_𝑗,𝑘. (66) In other words, the error term 𝑔_𝑗=𝐏_𝑗+1𝑓−𝐏_𝑗𝑓 between consecutive projections is given by the frame expansion: 𝑔_𝑗=^𝑁_{𝑖=1𝑘∈ℤ2}𝑓,𝜓^𝑖_𝑗,𝑘𝜓^𝑖_𝑗,𝑘. (67)

Suppose that the error term 𝑔_𝑗 has another expansion in terms of the frames {𝜓¹,…,𝜓^𝑁}, that is, 𝑔_𝑗=^𝑁_{𝑖=1𝑘∈ℤ2}𝑐_𝑗,𝑘𝜓^𝑖_𝑗,𝑘. (68) Then by using both (67) and (68), we have 𝑔_𝑗,𝑓=^𝑁_{𝑖=1𝑘∈ℤ2}𝑓,𝜓^𝑖_𝑗,𝑘²=^𝑁_{𝑖=1𝑘∈ℤ2}𝑐_𝑗,𝑘𝑓,𝜓^𝑖_𝑗,𝑘, (69) and this derives 0≤^𝑁_{𝑖=1𝑘∈ℤ2}𝑐_𝑗,𝑘−𝑓,𝜓^𝑖_𝑗,𝑘²=^𝑁_{𝑖=1𝑘∈ℤ2}𝑐_𝑗,𝑘²−2^𝑁_{𝑖=1𝑘∈ℤ2}𝑐_𝑗,𝑘𝑓,𝜓^𝑖_𝑗,𝑘+^𝑁_{𝑖=1𝑘∈ℤ2}𝑓,𝜓^𝑖_𝑗,𝑘²=^𝑁_{𝑖=1𝑘∈ℤ2}𝑐_𝑗,𝑘²−^𝑁_{𝑖=1𝑘∈ℤ2}𝑓,𝜓^𝑖_𝑗,𝑘². (70) This inequality means that the coefficients of the error term 𝑔_𝑗 in (67) have minimal 𝑙²-norm among all sequences {𝑐_𝑗,𝑘} which satisfy (68).

We next discuss minimum-energy bivariate wavelet frames decomposition and reconstruction. For any 𝑓∈𝐿²(ℝ²), we define the coefficients 𝑐_𝑗,𝑘∶=𝑓,𝜙_𝑗,𝑘,𝑑_𝑗,𝑘∶=𝑓,𝜓^𝑖_𝑗,𝑘𝑖=1,…,𝑁. (71) Then we can derive the decomposition and reconstruction formulas that are similar to those of orthonormal wavelets.

(1) Decomposition Algorithm. Suppose that coefficients {𝑐_𝑗+1,𝑙∶𝑙∈ℤ²} are known. By the two-scale relations (8) and (13), we have 𝜙_𝑗,𝑙(𝑥)=1√𝑠_𝑘∈ℤ2𝑝_{𝑘−𝐴𝑙}𝜙_𝑗+1,𝑘(𝑥),𝜓^𝑖_𝑗,𝑙(𝑥)=1√𝑠_𝑘∈ℤ2𝑞^𝑖_{𝑘−𝐴𝑙}𝜓^𝑖_𝑗+1,𝑘(𝑥),𝑖=1,…,𝑁. (72) Then, decomposition algorithm is given as 𝑐_𝑗,𝑙=1√𝑠_𝑘∈ℤ2𝑝_{𝑘−𝐴𝑙}𝑐_𝑗+1,𝑘,𝑑^𝑖_𝑗,𝑙=1√𝑠_𝑘∈ℤ2𝑞^𝑖_{𝑘−𝐴𝑙}𝑑^𝑖_𝑗+1,𝑘,𝑖=1,…,𝑁. (73)

(2) Reconstruction Algorithm. From (26), it follows that 𝜙_𝑗+1,𝑙(𝑥)=1√𝑠_𝑘∈ℤ2𝑝^∗_{𝑙−𝐴𝑘}𝜙_𝑗,𝑘(𝑥)+^𝑁_𝑖=1𝑞^𝑖∗_{𝑙−𝐴𝑘}𝜓^𝑖_𝑗,𝑘(𝑥). (74) Take the inner products on both sides of (74) with 𝑓, we get 𝑐_𝑗+1,𝑙=1√𝑠_𝑘∈ℤ2𝑝^∗_{𝑙−𝐴𝑘}𝑐_𝑗,𝑘+^𝑁_𝑖=1𝑞^𝑖∗_{𝑙−𝐴𝑘}𝑑^𝑖_𝑗,𝑘. (75)

5. Numerical Examples

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

Example 1. Let 𝐴=2𝐼₂; then 𝑠=4 and 𝜇₀=(0,0)^𝑇,𝜇₁=(1,0)^𝑇,𝜇₂=(0,1)^𝑇,𝜇₃=(1,1)^𝑇. (76)
Suppose that the Fourier transform 𝜙(𝜉) of scaling function 𝜙(𝜉) is 𝜙(𝜉)=1−𝑒^−𝑖𝜉1𝑖𝜉₁⋅1−𝑒^−𝑖𝜉2𝑖𝜉₂⋅1−𝑒^{−𝑖(𝜉1+𝜉2)}𝑖𝜉₁+𝜉₂. (77) Then its symbol function 𝑚₀(𝜉) satisfies 𝑚₀(𝜉)=181+𝑒^−𝑖𝜉11+𝑒^−𝑖𝜉21+𝑒^{−𝑖(𝜉1+𝜉2)}. (78) Thus, 𝑓⁰₀(𝑢)=141+𝑢^(1,1)𝑇,𝑓⁰₁(𝑢)=141+𝑢^(0,1)𝑇,𝑓⁰₂(𝑢)=141+𝑢^(1,0)𝑇,𝑓⁰₃(𝑢)=24. (79) Let 𝑓⁰₄(𝑢)=141−𝑢^(1,0)𝑇,𝑓⁰₅(𝑢)=141−𝑢^(0,1)𝑇,𝑓⁰₆(𝑢)=141−𝑢^(1,1)𝑇. (80) Then we have ∑⁶_𝑙=0|𝑓⁰_𝑖(𝑢)|²=1. By Corollary 9, there exists a minimum-energy bivariate wavelet frame associated with 𝜙(𝜉). Using Theorem 8, let 𝐻₁=√22−√2211111√22√22,𝐻₂=11√22−√221√22√2211,𝐻₃=1√22−√22111√22√221,𝐻₄=14√2√2√22√2√2√2002−2√2√22002−2√√20000000000−222√2002√20−2√√20022000−2−22√2−√2−2−√2√2√2,𝐷₁=diag𝑢^(−1,−1),𝐷,1,1,1,1,1,1₂=diag1,1,𝑢^(−1,0),𝐷,1,1,1,1₃=diag1,𝑢^(0,−1),1,1,1,1,1. (81) Thus, we have

𝐻^∗₁𝐷^∗₁𝐻^∗₂𝐷^∗₂𝐻^∗₃𝐷^∗₃𝐻^∗₄=14^{1+𝑢(1,1)02𝑢(1,1)√20−√2+2𝑢(1,1)1+𝑢(1,1)1+𝑢(0,1)0−2𝑢(0,1)√−20−√2+2𝑢(0,1)1+𝑢(0,1)1+𝑢(1,0)√√2+2𝑢(1,0)00−2+2𝑢(1,0)0−1−𝑢(1,0)√2−220000−2}_{1−𝑢(1,0)√√2−2𝑢(1,0)00−2−2𝑢(1,0)0−1+𝑢(1,0)1−𝑢(0,1)02𝑢(0,1)√−20−√2−2𝑢(0,1)1−𝑢(0,1)1−𝑢(1,1)0−2𝑢(1,1)√20−√2−2𝑢(1,1)1−𝑢(1,1)}. (82)

Consequently, we obtain symbol functions 𝑚₁(𝜉)=18𝑒^−𝑖𝜉2√2+√2𝑒^{−𝑖2𝜉1}−2√2𝑒^{−𝑖(𝜉1+𝜉2)},𝑚₂(𝜉)=182𝑒^{−𝑖2(𝜉1+𝜉2)}−2𝑒^−𝑖𝜉1𝑒^{−𝑖2𝜉1},𝑚₃(𝜉)=182−2𝑒^−𝑖𝜉1,𝑚₄(𝜉)=18𝑒^−𝑖𝜉2−2+2𝑒^{−𝑖2𝜉1},𝑚₅(𝜉)=18−√2+√2𝑒^{−𝑖(2𝜉1+2𝜉2)}+𝑒^−𝑖𝜉1−√2+√2𝑒^{−𝑖2𝜉2},𝑚₆(𝜉)=181+𝑒^{−𝑖(2𝜉1+2𝜉2)}+𝑒^−𝑖𝜉11+𝑒^{−𝑖2𝜉2}−𝑒^−𝑖𝜉21+𝑒^{−𝑖2𝜉1}−2𝑒^{−𝑖(𝜉1+𝜉2)}. (83) The functions generated by the previous symbol functions are minimum-energy bivariate wavelet frame functions associated with 𝜙(𝑥).

Example 2. Let 𝐴=3𝐼₂; then 𝑠=9 and 𝜇₀=(0,0)^𝑇,𝜇₁=(1,0)^𝑇,𝜇₂=(2,0)^𝑇,𝜇₃=(0,1)^𝑇,𝜇₄=(0,2)^𝑇,𝜇₅=(1,1)^𝑇,𝜇₆=(2,1)^𝑇,𝜇₇=(1,2)^𝑇,𝜇₈=(2,2)^𝑇. (84) Suppose that the Fourier transform 𝜙(𝜉) of scaling function 𝜙(𝜉) is 𝜙(𝜉)=1−𝑒^−𝑖𝜉1𝑖𝜉₁⋅1−𝑒^−𝑖𝜉2𝑖𝜉₂⋅1−𝑒^{−𝑖(𝜉1+𝜉2)}𝑖𝜉₁+𝜉₂. (85) Then its symbol function 𝑚₀(𝜉) satisfies 𝑚₀(𝜉)=1271+𝑒^−𝑖𝜉1+𝑒^{−𝑖2𝜉1}1+𝑒^−𝑖𝜉2+𝑒^{−𝑖2𝜉2}×1+𝑒^{−𝑖(𝜉1+𝜉2)}+𝑒^{−𝑖2(𝜉1+𝜉2)}. (86) Thus, 𝑓⁰₀(𝑢)=191+𝑢^(1,1)𝑇,𝑓⁰₁(𝑢)=191+𝑢^(0,1)𝑇+𝑢^(1,1)𝑇,𝑓⁰₂(𝑢)=191+2𝑢^(0,1)𝑇,𝑓⁰₃(𝑢)=191+𝑢^(1,0)𝑇+𝑢^(1,1)𝑇,𝑓⁰₄(𝑢)=191+2𝑢^(1,0)𝑇,𝑓⁰₅(𝑢)=192+𝑢^(1,1)𝑇,𝑓⁰₆(𝑢)=192+𝑢^(0,1)𝑇,𝑓⁰₇(𝑢)=192+𝑢^(1,0)𝑇,𝑓⁰₈(𝑢)=39. (87) Let 𝑓⁰₉(𝑢)=√691−𝑢^(1,0)𝑇,𝑓⁰₁₀(𝑢)=√691−𝑢^(0,1)𝑇,𝑓⁰₁₁(𝑢)=√691−𝑢^(1,1)𝑇. (88) Then we have ∑¹¹_𝑙=0|𝑓⁰_𝑙(𝑢)|²=1. By Corollary 9, there exists a minimum-energy bivariate wavelet frame associated with 𝜙(𝜉).

Example 3. Let 𝐴=¹¹₁₋₁; then 𝑠=2 and 𝜇₀=(0,0)^𝑇,𝜇₁=(1,0)^𝑇. (89)
Suppose that the Fourier transform 𝜙(𝜉) of scaling function 𝜙(𝜉) satisfies 𝜙(𝜉)=1−𝑒^−𝑖𝜉1𝑖𝜉₁⋅1−𝑒^−𝑖𝜉2𝑖𝜉₂⋅1−𝑒^{−𝑖(𝜉1+𝜉2)}𝑖𝜉₁+𝜉₂⋅1−𝑒^{−𝑖(𝜉1−𝜉2)}𝑖𝜉₁−𝜉₂. (90) Then its symbol function 𝑚₀(𝜉) satisfies 𝑚₀(𝜉)=141+𝑒^−𝑖𝜉11+𝑒^−𝑖𝜉2. (91) Let 𝑧=𝑒^{−𝑖𝜉𝑇}=𝑒^−𝑖𝜉1𝑒^{−𝑖𝜉2𝑇}, and we have 𝑧^𝜇0=1,𝑧^𝜇1=𝑒^−𝑖𝜉1, 𝑢=𝑧^𝐴=(𝑒^{−𝑖(𝜉1+𝜉2)},𝑒^{−𝑖(𝜉1−𝜉2)}). Therefore, 𝑓⁰₀(𝑢)=12√21+𝑢^(1,0)𝑇,𝑓⁰₁(𝑢)=12√21+𝑢^(0,−1)𝑇. (92) Let 𝑓⁰₂(𝑢)=12√21−𝑢^(1,0)𝑇,𝑓⁰₃(𝑢)=12√21−𝑢^(0,−1)𝑇, (93) and we have ∑³_𝑙=0|𝑓⁰_𝑙(𝑢)|²=1. By Corollary 9, there exists a minimum-energy bivariate wavelet frame associated with 𝜙(𝜉).

6. Conclusions

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.

Acknowledgments

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.