Minimum-Energy Bivariate Wavelet Frame with Arbitrary Dilation Matrix

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 896050, 10 pages http://dx.doi.org/10.1155/2013/896050

Research Article Minimum-Energy Bivariate Wavelet Frame with Arbitrary Dilation Matrix Fengjuan Zhu, Qiufu Li, and Yongdong Huang School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan 750021, China Correspondence should be addressed to Yongdong Huang; [email protected] 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.

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 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 [4–18], 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.

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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 {𝜇𝑖 , 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)|2 𝑑𝑥 < ∞} and 𝑙2 (Z2 ) = {𝑠 : ∑𝑚∈Z2 |𝑠𝑚 |2 𝑑𝑥 < ∞}. For any 𝑓, 𝑔 ∈ 𝐿2 (R2 ), the inner product, norm, and the Fourier transform are defined, respectively, by 󵄩󵄩2

󵄩󵄩 󵄩󵄩𝑓󵄩󵄩 = ⟨𝑓, 𝑓⟩ ,

⟨𝑓 (𝑥) , 𝑔 (𝑥)⟩ = ∫ 𝑓 (𝑥) 𝑔 (𝑥)𝑑𝑥, R2

𝑓̂ (𝜉) =

1 ∫ 𝑓 (𝑥) 𝑒−𝑖𝑥⋅𝜉 𝑑𝑥. √2𝜋 R2

(1)

Definition 2 (see [18]). Let H be a complex and separable Hilbert space. A sequence {𝑓𝑘 : 𝑘 ∈ Z} is a frame for H if there exist constants 0 < 𝐴, 𝐵 < ∞, such that, for any 𝑓 ∈ H, 󵄩 󵄩2 󵄨 󵄨2 󵄩 󵄩2 𝐴󵄩󵄩󵄩𝑓󵄩󵄩󵄩 ≤ ∑ 󵄨󵄨󵄨⟨𝑓, 𝑓𝑘 ⟩󵄨󵄨󵄨 ≤ 𝐵󵄩󵄩󵄩𝑓󵄩󵄩󵄩 . 𝑘∈Z

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 = ∑ ⟨𝑓, 𝑓𝑘 ⟩ 𝑓𝑘 ,

𝑗 ∈ Z, 𝑘 ∈ Z2 .

Definition 3 (see [18]). A bivariate frame multiresolution analysis (FMRA) for 𝐿2 (R2 ) consists of a sequence of closed subspaces {𝑉𝑗 }𝑗∈Z and a function 𝜙 ∈ 𝑉0 such that (1) 𝑉𝑗 ⊂ 𝑉𝑗+1 , 𝑗 ∈ Z; (2) ⋂𝑗∈Z 𝑉𝑗 = {0}, ⋃𝑗∈Z 𝑉𝑗 = 𝐿2 (R2 ); (3) 𝑓(𝑥) ∈ 𝑉𝑗 ⇔ 𝑓(𝐴𝑥) ∈ 𝑉𝑗+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) 𝑘∈Z2

(2)

(5) For any 𝑧 = (𝑧1 , 𝑧2 )𝑇 , 𝑧1 ≠ 0, 𝑧2 ≠ 0, 𝑘 = (𝑘1 , 𝑘2 )𝑇 ∈ R2 , 𝑧−1 , 𝑧𝑘 are defined, respectively, by 𝑇

𝑧−1 = (𝑧1−1 , 𝑧2−1 ) ,

2

𝑘

𝑧𝑘 = ∏𝑧𝑖 𝑖 .

(4)

𝑖=1

(6) For any 𝐴 ∈ R2×2 , 𝑧𝐴 is defined by 𝑇

𝑧𝐴 = (𝑧𝐴 1 , 𝑧𝐴 2 ) , where 𝐴 𝑖 is the 𝑖-th column of 𝐴. Now, we give the definition of frame.

(5)

(8)

where {𝑝𝑘 }𝑘∈Z2 ∈ 𝑙2 (Z2 ). The Fourier transform of (8) is 𝜙̂ (𝜉) = 𝑚0 (𝐴−𝑇 𝜉) 𝜙̂ (𝐴−𝑇 𝜉) ,

𝜉 ∈ R2 ,

(9)

where 𝑚0 (𝜉) =

(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.

(7)

2.2. Minimum-Energy Bivariate Wavelet Frame

𝜙 (𝑥) = ∑ 𝑝𝑘 𝜙 (𝐴𝑥 − 𝑘) ,

(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 ) ;

∀𝑓 ∈ H.

𝑘∈Z

(3) For any function 𝑓 ∈ 𝐿2 (R2 ), 𝑓𝑗,𝑘 (𝑥) is defined by 𝑓𝑗,𝑘 (𝑥) = 𝑠𝑗/2 𝑓 (𝐴𝑗 𝑥 − 𝑘) ,

(6)

𝑇 1 1 ∑ 𝑝𝑘 𝑒−𝑖𝜉 𝑘 = ∑ 𝑝𝑘 𝑧𝑘 , 𝑠 𝑘∈Z2 |det (𝐴)| 𝑘∈Z2

(10)

where 𝑚0 (𝜉) is the symbol function of scaling function 𝜙(𝑥), 𝑇 𝑧 = 𝑒−𝑖𝜉 . 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 𝜙̂ ∈ 𝐿∞ and 𝜙̂ ̂ continuous at 0 and 𝜙(0) = 1. Suppose that 𝜙 generates a sequence of nested closed subspaces {𝑉𝑗 }𝑗∈Z , then Ψ = {𝜓1 , 𝜓2 , . . . , 𝜓𝑁} ⊂ 𝑉1 is called a minimum-energy bivariate wavelet frame associated with 𝜙 if for all 𝑓 ∈ 𝐿2 (R2 ): 𝑁 󵄨 󵄨2 󵄨 󵄨 󵄨2 󵄨2 𝑙 ⟩󵄨󵄨󵄨󵄨 . ∑ 󵄨󵄨󵄨⟨𝑓, 𝜙1,𝑘 ⟩󵄨󵄨󵄨 = ∑ 󵄨󵄨󵄨⟨𝑓, 𝜙0,𝑘 ⟩󵄨󵄨󵄨 + ∑ ∑ 󵄨󵄨󵄨󵄨⟨𝑓, 𝜓0,𝑘 2 2 2

𝑘∈Z

𝑘∈Z

𝑙=1 𝑘∈Z

(11)

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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 ∑ ⟨𝑓, 𝜙1,𝑘 ⟩ 𝜙1,𝑘 = ∑ ⟨𝑓, 𝜙0,𝑘 ⟩ 𝜙0,𝑘

𝑘∈Z2

𝑘∈Z2

𝑁

+∑ ∑ 𝑙=1 𝑘∈Z2

𝑙 𝑙 ⟨𝑓, 𝜓0,𝑘 ⟩ 𝜓0,𝑘 ,

2

Theorem 6. Suppose that the symbols 𝑚𝑙 (𝜉), 𝑙 = 0, 1, . . . , 𝑁 in (10) and (14) are Laurent polynomial, and generate the refinable function 𝜙(𝑥) and Ψ = {𝜓1 , 𝜓2 , . . . , 𝜓𝑁}. If 𝜙̂ is ̂ = 1 and 𝜙(𝑥) generates a nested closed continuous at 0 and 𝜙(0) subspaces sequence {𝑉𝑗 }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)

2

∀𝑓 ∈ 𝐿 (R ) . (12)

𝑀 (𝜉) 𝑀∗ (𝜉) = 𝐼𝑠 ,

The interpretation of minimum-energy bivariate wavelet frame will be clarified later. Consider Ψ = {𝜓1 , 𝜓2 , . . . , 𝜓𝑁} ⊂ 𝑉1 , with 𝑙

𝜓 (𝑥) = ∑ 𝑘∈Z2

𝑞𝑘𝑙 𝜙 (𝐴𝑥

− 𝑘) ,

𝑙 = 1, 2, . . . , 𝑁.

𝑇 1 1 𝑚𝑙 (𝜉) = ∑ 𝑞𝑘𝑙 𝑒−𝑖𝜉 𝑘 = ∑ 𝑞𝑘𝑙 𝑧𝑘 , 𝑠 𝑘∈Z2 𝑠 𝑘∈Z2

𝑙 = 1, 2, . . . , 𝑁, (14)

𝑁

𝑖=1

𝑘∈Z2

where 𝑧 = 𝑒 . With 𝑚0 (𝜉), 𝑚1 (𝜉), . . . , 𝑚𝑁(𝜉), we formulate the 𝑠×(𝑁+1) matrix 𝑀(𝜉):

∀𝑚, 𝑙 ∈ Z2 .

− 𝑠𝛿𝑚𝑙 = 0,

Proof. By using the two-scale relations (8) and (13) and notation 𝛼𝑚𝑙 , then formula (12) is equivalent to ∑ 𝛼𝑚𝑙 ⟨𝑓, 𝜙 (𝐴𝑥 − 𝑚)⟩ 𝜙 (𝐴𝑥 − 𝑙) = 0,

∀𝑓 ∈ 𝐿2 (R2 ) .

𝑚,𝑙∈Z2

On the other hand, formula (17) can be reformulated as 𝑁

󵄨2 󵄨 ∑󵄨󵄨󵄨𝑚𝑙 (𝜉)󵄨󵄨󵄨 = 1, 𝑙=0

𝑀 (𝜉) 𝑚0 (𝜉) 𝑚1 (𝜉) ⋅⋅⋅ 𝑚𝑁 (𝜉) 𝑚0 (𝜉 + 2𝐴−𝑇 𝜇1 𝜋) 𝑚1 (𝜉 + 2𝐴−𝑇 𝜇1 𝜋) ⋅ ⋅ ⋅ 𝑚𝑁 (𝜉 + 2𝐴−𝑇 𝜇1 𝜋) . . . ( ), . . . . . . −𝑇 −𝑇 −𝑇 𝑚0 (𝜉 + 2𝐴 𝜇𝑠−1 𝜋) 𝑚1 (𝜉 + 2𝐴 𝜇𝑠−1 𝜋) ⋅ ⋅ ⋅ 𝑚𝑁 (𝜉 + 2𝐴 𝜇𝑠−1 𝜋)

∑𝑚𝑙 (𝜉) 𝑚𝑙∗ (𝜉 + 2𝐴−1 𝜇𝑗 𝜋) = 0,

𝑗 = 1, . . . , 𝑠 − 1

𝑙=0

which is equivalent to 𝑁

𝑠−1

𝑙=0

𝑗=0

∑𝑚𝑙 (𝜉) ∑ 𝑚𝑙∗ (𝜉 + 2𝐴−1 𝜇𝑗 𝜋) = 1,

and 𝑀∗ (𝜉) denotes the complex conjugate of the transpose of 𝑀(𝜉). 𝑁

𝑠−1

𝑙=0

𝑗=1

∑𝑚𝑙 (𝜉) (𝑚0∗ (𝜉) − ∑ 𝑚𝑙∗ (𝜉 + 2𝐴−1 𝜇𝑗 𝜋)) = 1,

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 −1

𝑒𝑖2𝜋𝜇1 𝐴 𝜇1 𝑇 −1 𝑒𝑖2𝜋𝜇2 𝐴 𝜇1 A=( .. . 𝑇

(20)

𝑁

(15)

𝑇

(18)

(19)

−𝑖𝜉𝑇

=

(17)

∗ 𝑖∗ 𝑖 𝑝𝑙−𝐴𝑘 + ∑𝑞𝑚−𝐴𝑘 𝑞𝑙−𝐴𝑘 ) 𝛼𝑚𝑙 = ∑ (𝑝𝑚−𝐴𝑘

(3)

(13)

Using Fourier transform on the previous equation, we can get their symbols as follows:

∀𝜉 ∈ R2 .

−1

𝑒𝑖2𝜋𝜇𝑠−1 𝐴

𝜇1

𝑇

−1

𝑇

𝑁

𝑠−1

𝑙=0

𝑗=0

∑𝑚𝑙 (𝜉) ( ∑ 𝑚𝑙∗ (𝜉+2𝐴−1 𝜇𝑗 𝜋) − 2𝑚𝑙 (𝜉 + 2𝐴−1 𝜇𝑛 𝜋)) = 1, 𝑛 = 1, . . . , 𝑠 − 1; 𝜉 ∈ R2 , (21) or

−1

𝑒𝑖2𝜋𝜇1 𝐴 𝜇2 ⋅ ⋅ ⋅ 𝑒𝑖2𝜋𝜇1 𝐴 𝜇𝑠−1 𝑇 −1 𝑇 −1 𝑒𝑖2𝜋𝜇2 𝐴 𝜇2 ⋅ ⋅ ⋅ 𝑒𝑖2𝜋𝜇2 𝐴 𝜇𝑠−1 ); .. .. . . 𝑇

−1

𝑒𝑖2𝜋𝜇𝑠−1 𝐴

𝜇2

𝑇

−1

⋅ ⋅ ⋅ 𝑒𝑖2𝜋𝜇𝑠−1 𝐴

𝑁

𝑙∗ 𝑧𝐴𝑘 = 1, ∑𝑚𝑙 (𝜉) ∑ 𝑞−𝐴𝑘 𝑙=0

𝜇𝑠−1

The following theorem presents the equivalent characterizations of the minimum-energy bivariate wavelet frames with arbitrary dilation matrix.

𝑁

𝑠−1

𝑙=0

𝑛=1𝑘∈Z2

∑𝑚𝑙 (𝜉) ( ∑ ∑ 𝑞𝜇𝑙∗𝑛 −𝐴𝑘 𝑧𝐴𝑘−𝜇𝑛 ) = 𝑠 − 1,

(16) then, A is invertible matrix.

𝑘∈Z2

𝑁

𝑠−1

𝑙=0

𝑛=1

𝑇

−1

∑𝑚𝑙 (𝜉) ( ∑ 𝑒𝑖2𝜇𝑗 𝐴

𝜇𝑛 𝜋

∑ 𝑞𝜇𝑙∗𝑛 −𝐴𝑘 𝑧𝐴𝑘−𝜇𝑛 ) = −1,

𝑘∈Z2

𝑗 = 1, . . . , 𝑠 − 1; 𝜉 ∈ R2 ,

(22)

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where 𝑞𝑘0 = 𝑝𝑘 , 𝑘 ∈ Z2 . Since A is an invertible matrix, the previous equation is equivalent to 𝑁

∑𝑚𝑙 (𝜉) ∑ 𝑞𝜇𝑙∗𝑛 −𝐴𝑘 𝑧𝐴𝑘−𝜇𝑛 = 1, 𝑙=0

𝑛 = 0, 1, . . . , 𝑠 − 1.

(23)

Theorem 7. A compactly supported refinable function 𝜙 ∈ ̂ = 1, and 𝑚0 (𝜉) is 𝐿2 (R2 ), with 𝜙̂ continuous at 0 and 𝜙(0) the symbol function of 𝜙. Let Ψ = {𝜓1 , 𝜓2 , . . . , 𝜓𝑁} be the minimum-energy multiwavelet frames associated with 𝜙; then 𝑠−1

𝑘∈Z2

̂ −𝑇 𝜉), 𝑛 = We multiply the identities in (23) by 𝑧 𝜙(𝐴 −𝑖𝜉𝑇 𝐴−1 0, 1, . . . , 𝑠 − 1, respectively, where 𝑧 = 𝑒 , and we get

󵄨 󵄨2 ∑ 󵄨󵄨󵄨󵄨𝑚0 (𝜉 + 2𝐴−1 𝜇𝑛 𝜋)󵄨󵄨󵄨󵄨 ≤ 1,

𝜇𝑛

𝑁

𝜙̂ (𝐴−𝑇 𝜉) 𝑧𝜇𝑛 = ∑ ∑ 𝑞𝜇𝑙∗𝑛 −𝐴𝑘 𝑧𝐴𝑘 𝜓̂𝑙 (𝜉) ,

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) = 𝐼𝑠 .

(30)

∗

(24) let 𝜓0 = 𝜙. Take the Fourier transform on the two sides of the previous formula, (23) is equivalent to 𝑁

𝑠𝜙 (𝐴𝑥 − 𝜇𝑛 ) = ∑ ∑ 𝑞𝜇𝑙∗𝑛 −𝐴𝑘 𝜓𝑙 (𝑥 − 𝑘) ,

Since 𝑄(𝜉)𝑄 (𝜉) is a Hermitian matrix, the matrix 𝐼𝑠 − 𝐷(𝜉)𝐷∗ (𝜉) is positive semidefinite. We have 𝐷 (𝜉) −𝐷 (𝜉) 𝐼 𝐼 ( ∗𝑠 ) ( ∗𝑠 ) 𝐷 (𝜉) 1 −𝐷 (𝜉) 1 𝐼 − 𝐷 (𝜉) 𝐷∗ (𝜉) =( 𝑠

𝑛 = 0, 1, . . . , 𝑠 − 1.

𝑙=0 𝑘∈Z2

)

1 − 𝐷∗ (𝜉) 𝐷 (𝜉)

(25) 𝐷 (𝜉) 𝐷 (𝜉) 𝐼 𝐼 ) = det ( 𝑠 ), det ( ∗ 𝑠 𝐷 (𝜉) 1 1 − 𝐷∗ (𝜉) 𝐷 (𝜉)

Thus, 𝑁

𝑙∗ 𝑠𝜙 (𝐴𝑥 − 𝜇) = ∑ ∑ 𝑞𝜇−𝐴𝑘 𝜓𝑙 (𝑥 − 𝑘) , 𝑙=0

∀𝜇 ∈ Z2 .

(26)

𝑘∈Z2

By using the two-scale relations (8) and (13), we can rewrite formula (26) as ∑ 𝛼𝑚𝑙 𝜙 (𝐴𝑥 − 𝑚) = 0,

(29)

𝑛=0

𝑛 = 0, 1, . . . , 𝑠 − 1;

𝑙=0 𝑘∈Z2

∀𝜉 ∈ R2 .

2

∀𝑙 ∈ Z .

𝑚∈Z2

𝑙 ∈ Z2 .

−𝐷 (𝜉) −𝐷 (𝜉) 𝐼𝑠 𝐼 ). ) = det ( 𝑠 −𝐷∗ (𝜉) 1 − 𝐷∗ (𝜉) 𝐷 (𝜉) 1

Therefore, det (𝐼𝑠 − 𝐷 (𝜉) 𝐷∗ (𝜉)) (1 − 𝐷∗ (𝜉) 𝐷 (𝜉))

(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 𝛽𝑙 (𝑓) = ⟨𝑓, ∑ 𝛼𝑚𝑙 𝜙 (𝐴𝑥 − 𝑚)⟩ ,

det (

(28)

𝑚∈Z2

Then by using the properties that, for every fixed 𝑚, 𝛼𝑚𝑙 = 0 except for finitely many 𝑙, and both 𝜙 and 𝑓 have compact support, it is clear that only finitely many of the values 𝛽𝑙 (𝑓) ̂ is a nontrivial function, by taking is nonzero. Now, since 𝜙(𝜉) the Fourier transform of (19), it follows that the trigonometric 𝑇 −𝑇 polynomial ∑𝑙∈Z2 𝛽𝑙 (𝑓)𝑒−𝑖𝜉 𝐴 𝑙 ≡ 0. Obviously, 𝛽𝑙 (𝑓) = 0, 𝑙 ∈ Z2 . By choosing 𝑓 = ∑𝑚∈Z2 𝛼𝑚𝑙 𝜙(𝐴𝑥 − 𝑚), we get formula (27). By taking the Fourier transform of (27), we get 𝛼𝑚𝑙 = 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.

(31)

= (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), . . . , 𝑚𝑁(𝑧), we give two sufficient conditions. 𝑇 Let 𝑧 = 𝑒−𝑖𝜉 ; then 𝑚0 (𝜉) =

𝑇 𝑇 1 𝑠−1 1 ∑ 𝑝𝑘 𝑒−𝑖𝜉 𝑘 = ∑ ∑ 𝑝𝜇𝑖 +𝐴𝑘 𝑒−𝑖𝜉 (𝜇𝑖 +𝐴𝑘) 𝑠 𝑘∈Z2 𝑠 𝑖=0 𝑘∈Z2

=

𝑇 1 𝑠−1 −𝑖𝜉𝑇 𝜇𝑖 ∑𝑒 ∑ 𝑝𝜇𝑖 +𝐴𝑘 𝑒−𝑖𝜉 𝐴𝑘 𝑠 𝑖=0 𝑘∈Z2

=

1 𝑠−1 𝜇𝑖 1 𝑠−1 𝜇𝑖 0 𝐴 ∑𝑧 ∑ 𝑝𝜇𝑖 +𝐴𝑘 𝑧𝐴𝑘 = ∑𝑧 𝑓𝑖 (𝑧 ) , √𝑠 𝑖=0 𝑠 𝑖=0 𝑘∈Z2

(34)

where 𝑓𝑖0 (𝑥) = (1/√𝑠) ∑𝑘∈Z2 𝑝𝜇𝑖 +𝐴𝑘 𝑥𝑘 , 𝑥 ∈ R2 . Similarly, 𝑚𝑙 (𝜉) =

1 𝑠−1 𝜇𝑖 𝑙 𝐴 ∑𝑧 𝑓𝑖 (𝑧 ) , √𝑠 𝑖=0

𝑙 = 1, 2, . . . , 𝑁,

(35)

Journal of Applied Mathematics

5

where 𝑓𝑖𝑙 (𝑥) = (1/√𝑠) ∑𝑘∈Z2 𝑞𝜇𝑙 𝑖 +𝐴𝑘 𝑥𝑘 , 𝑥 ∈ R2 . Let 𝐶 (𝑧) =

Proof. Under the assumption, we know that the vector

1 √𝑠

0 f := [𝑓00 (𝑢) , 𝑓10 (𝑢) , . . . , 𝑓𝑠−1 (𝑢) , 𝑓𝑠0 (𝑢)] 𝑇 −1

𝑇 −1

𝑇 −1

𝑧𝜇0 𝑒−𝑖2𝜋𝜇0 𝐴 𝜇0 𝑧𝜇1 𝑒−𝑖2𝜋𝜇0 𝐴 𝜇1 ⋅ ⋅ ⋅ 𝑧𝜇𝑠−1 𝑒−𝑖2𝜋𝜇0 𝐴 𝜇𝑠−1 𝑇 −1 𝑇 −1 𝑇 −1 𝑧𝜇0 𝑒−𝑖2𝜋𝜇1 𝐴 𝜇0 𝑧𝜇1 𝑒−𝑖2𝜋𝜇1 𝐴 𝜇1 ⋅ ⋅ ⋅ 𝑧𝜇𝑠−1 𝑒−𝑖2𝜋𝜇1 𝐴 𝜇𝑠−1 ); ×( .. .. .. . . . 𝑇 −1 𝑇 −1 𝑇 −1 𝑧𝜇0 𝑒−𝑖2𝜋𝜇𝑠−1 𝐴 𝜇0 𝑧𝜇1 𝑒−𝑖2𝜋𝜇𝑠−1 𝐴 𝜇1 ⋅ ⋅ ⋅ 𝑧𝜇3 𝑒−𝑖2𝜋𝜇𝑠−1 𝐴 𝜇𝑠−1

f1 = 𝐷0 f = [𝑢𝑡0 𝑓00 (𝑢) , 𝑢𝑡1 𝑓10 (𝑢) , . . . , 𝑢𝑡𝑠 𝑓𝑠0 (𝑢)] 𝐾

= ∑ a𝑗 𝑢 𝑗 ,

thus,

𝑓00 (𝑧𝐴) 𝑓01 (𝑧𝐴) ⋅ ⋅ ⋅ 𝑓0𝑁 (𝑧𝐴 ) 𝑓10 (𝑧𝐴) 𝑓11 (𝑧𝐴) ⋅ ⋅ ⋅ 𝑓1𝑁 (𝑧𝐴 ) ( ) .. .. .. . . . 0 1 𝑁 𝑓𝑠−1 (𝑧𝐴) 𝑓𝑠−1 (𝑧𝐴) ⋅ ⋅ ⋅ 𝑓𝑠−1 (𝑧𝐴) ∗

For convenience, let 𝑢 = 𝑧𝐴 , condition (29) can be 0 2 rewritten as ∑𝑠−1 𝑛=0 |𝑓𝑛 (𝑢)| ≤ 1. 0 If there exists 𝑓𝑠 (𝑢) such that 𝑠

󵄨2 󵄨 ∑ 󵄨󵄨󵄨󵄨𝑓𝑛0 (𝑢)󵄨󵄨󵄨󵄨 = 1,

(39)

𝑛=0

then, we have the following Theorem 8. Theorem 8. Let 𝜙(𝑥) ∈ 𝐿2 (R2 ) be a compactly supported ̂ refinable function, with 𝜙̂ continuous at 0 and 𝜙(0) = 1, and its symbol function satisfies 𝑠−1

󵄨2 󵄨 ∑ 󵄨󵄨󵄨󵄨𝑓𝑛0 (𝑢)󵄨󵄨󵄨󵄨 ≤ 1.

(40)

𝑛=0

If there exists 𝑓𝑠0 (𝑢) such that 𝑠

󵄨2 󵄨 ∑ 󵄨󵄨󵄨󵄨𝑓𝑛0 (𝑢)󵄨󵄨󵄨󵄨 = 1,

(41)

𝑛=0

then there exists a minimum-energy bivariate wavelet frame associated with 𝜙(𝑥).

(43)

where a𝑗 ∈ R𝑠+1 with a0 ≠ 0 and a𝐾 ≠ 0. It is clear that f1 is a unit vector: f1∗ f1

∗

𝐾

𝐾

𝑗

= ( ∑ a𝑗 𝑢 ) ( ∑ a𝑗 𝑢𝑗 ) = 1, 𝑗=(0,0)

∀ |𝑢| = 1, (44)

𝑗=(0,0)

and consequently a𝑇0 a𝐾 = 0. We next consider the (𝑠 + 1) × (𝑠 + 1) Householder matrix: 𝐻1 = 𝐼𝑠+1 −

𝑓00 (𝑧𝐴 ) 𝑓01 (𝑧𝐴 ) ⋅ ⋅ ⋅ 𝑓0𝑁 (𝑧𝐴) 𝑓10 (𝑧𝐴 ) 𝑓11 (𝑧𝐴 ) ⋅ ⋅ ⋅ 𝑓1𝑁 (𝑧𝐴) ×( ) = 𝐼𝑠 . .. .. .. . . . 0 1 𝑁 𝑓𝑠−1 (𝑧𝐴 ) 𝑓𝑠−1 (𝑧𝐴 ) ⋅ ⋅ ⋅ 𝑓𝑠−1 (𝑧𝐴 ) (38)

𝑇

𝑡0 , 𝑡1 , . . . , 𝑡𝑠 ∈ Z2 ,

𝑗=(0,0)

Since 𝐶(𝑧) is a unitary matrix, condition (17) is equivalent to

(42)

is a unit vector. Construct diagonal matrix 𝐷0 = diag(𝑢𝑡0 , 𝑢𝑡1 , . . . , 𝑢𝑡𝑠 ) such that

(36)

𝑓00 (𝑧𝐴 ) 𝑓01 (𝑧𝐴 ) ⋅ ⋅ ⋅ 𝑓0𝑁 (𝑧𝐴) 𝑓10 (𝑧𝐴 ) 𝑓11 (𝑧𝐴 ) ⋅ ⋅ ⋅ 𝑓1𝑁 (𝑧𝐴) 𝑀 (𝜉) = 𝐶 (𝑧) ( ). .. .. .. . . . 0 1 𝑁 𝑓𝑠−1 (𝑧𝐴 ) 𝑓𝑠−1 (𝑧𝐴 ) ⋅ ⋅ ⋅ 𝑓𝑠−1 (𝑧𝐴) (37)

𝑇

2 kk𝑇, ‖k‖2

(45)

where k = a𝐾 ± ‖a𝐾 ‖e1 , with e1 = (1, 0, . . . , 0)𝑇𝑠+1 , and the + or − signs are so chosen that k ≠ 0. Then 󵄩 󵄩 𝐻1 a𝐾 = ± 󵄩󵄩󵄩a𝐾 󵄩󵄩󵄩 e1 . (46) Since Householder matrix is orthogonal matrix, we have 𝑇

(𝐻1 a0 ) (𝐻1 a𝐾 ) = a𝑇0 𝐻1𝑇 𝐻1 a𝐾 = a𝑇0 a𝐾 = 0.

(47)

By the previous equation, the first component of 𝐻1 a0 is 0. 𝑗 Now 𝐻1 f1 = ∑𝐾 𝑗=(0,0) (𝐻1 a𝑗 )𝑢 , we construct diagonal matrix 𝑡(1) 𝐷1 = diag(𝑢 , 1, . . . , 1), 𝑡(1) ∈ Z2 such that 𝐾

𝐾

𝑗=(0,0)

𝑗=(0,0)

𝑗 f2 = 𝐷1 𝐻1 f1 = 𝐷1 ∑ (𝐻1 a𝑗 ) 𝑢𝑗 = ∑ a(1) 𝑗 𝑢

(48)

(1) is also a unit vector and 𝐾1 < 𝐾, a(1) 0 ≠ 0, a𝐾1 ≠ 0. Similarly, we define the Householder matrix:

𝐻2 = 𝐼𝑠+1 −

2 k k𝑇 , ‖k‖2

(49)

where k = a𝐾1 ± ‖a𝐾1 ‖e1 ≠ 0, and 𝐷2 = diag(𝑢𝑡(2) , 1, . . . , 1) such that 𝐾1

𝐾1

𝑗=(0,0)

𝑗=(0,0)

𝑗 (2) 𝑗 f3 = 𝐷2 𝐻2 f2 = 𝐷2 ∑ (𝐻2 a(1) 𝑗 ) 𝑢 = ∑ a𝑗 𝑢

(50)

(2) is also a unit vector and 𝐾2 < 𝐾1 , a(2) 0 ≠ 0, a𝐾2 ≠ 0. Since every component of f is a finite sum, we repeat this procedure finite times to get some unitary matrices 𝐷𝐿 , 𝐻𝐿 , 𝐷𝐿−1 , 𝐻𝐿−1 , . . . , 𝐻2 , 𝐷1 , 𝐻1 ,

𝐷𝐿 𝐻𝐿 𝐷𝐿−1 𝐻𝐿−1 ⋅ ⋅ ⋅ 𝐻2 𝐷1 𝐻1 𝐷0 f = e1 .

(51)

6

Journal of Applied Mathematics

That is, f is the first column of the unitary matrix ∗ ∗ 𝐻 := 𝐷0∗ 𝐻0∗ 𝐷1∗ 𝐻2∗ ⋅ ⋅ ⋅ 𝐻𝐿−1 𝐷𝐿−1 𝐻𝐿∗ 𝐷𝐿∗ .

𝛽𝑘 =

1 ( − 𝑚0 (𝜉 + 2𝐴−𝑇 𝜇0 𝜋) 𝑚0∗ (𝜉 + 2𝐴−𝑇 𝜇𝑘 𝜋) , Ω𝑘

(52)

− 𝑚0 (𝜉 + 2𝐴−𝑇 𝜇1 𝜋) 𝑚0∗ (𝜉 + 2𝐴−𝑇 𝜇𝑘 𝜋) , . . . , − 𝑚0 (𝜉 + 2𝐴−𝑇 𝜇𝑘−1 𝜋) 𝑚0∗ (𝜉 + 2𝐴−𝑇 𝜇𝑘 𝜋) ,

Let

𝑘−1

𝑓00 (𝑢) 𝑓01 (𝑢) ⋅ ⋅ ⋅ 𝑓0𝑠 (𝑢) 0 1 𝑠 𝐻 = (𝑓1 (𝑢) 𝑓1 (𝑢) ⋅ ⋅ ⋅ 𝑓1 (𝑢)) ; .. .. .. . . . 0 1 𝑠 𝑓𝑠 (𝑢) 𝑓𝑠 (𝑢) ⋅ ⋅ ⋅ 𝑓𝑠 (𝑢)

󵄨 󵄨2 ∑ 󵄨󵄨󵄨󵄨𝑚0 (𝜉 + 2𝐴−𝑇 𝜇𝑗 𝜋)󵄨󵄨󵄨󵄨 , 0, . . . , 0)

𝑗=0

(53)

𝑘 = 2, 3, . . . , 𝑠 − 1, 𝛽𝑠 =

1 (𝑚0 (𝜉 + 2𝐴−𝑇 𝜇0 𝜋) , Ω𝑠

then, 𝐻 satisfies (38). In the formula (15), we let 1 𝑠−1 𝜇𝑖 𝑙 𝐴 𝑚𝑙 (𝜉) = ∑𝑧 𝑓𝑖 (𝑧 ) , √𝑠 𝑖=0

𝑇

𝑚0 (𝜉 + 2𝐴−𝑇 𝜇1 𝜋) , . . . , 𝑚0 (𝜉 + 2𝐴−𝑇 𝜇𝑠−1 𝜋))

(58) 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 𝜙̂ continuous at 0 and 𝜙(0) = 1, and its symbol function satisfies 𝑠−1

󵄨 󵄨2 ∑ 󵄨󵄨󵄨󵄨𝑚0 (𝜉 + 2𝐴−1 𝜇𝑛 𝜋)󵄨󵄨󵄨󵄨 ≤ 1,

∀𝜉 ∈ R2 .

(55)

𝑛=0

Λ (𝜉) = diag (1, . . . , 1, 𝜆 𝑠 ) , (59)

then, 𝐼𝑠 − 𝐷 (𝜉) 𝐷∗ (𝜉) = 𝑆 (𝜉) Λ (𝜉) 𝑆∗ (𝜉) . 2

(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 (57)

Then

Theorem 10. Let 𝜙(𝑥) ∈ 𝐿 (R ) be a compactly supported ̂ refinable function, with 𝜙̂ continuous at 0 and 𝜙(0) = 1, and its symbol function satisfies 𝑠−1

∀𝜉 ∈ R2 .

(61)

𝑠 󵄨2 󵄨 ∑󵄨󵄨󵄨󵄨𝑓𝑖0 (𝑢)󵄨󵄨󵄨󵄨 = 1,

(62)

𝑖=0

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 𝑓𝑠0 (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))

𝜆 1 = ⋅ ⋅ ⋅ = 𝜆 𝑠−1 = 1,

(60)

2

If there exists 𝑓𝑠0 (𝑢) satisfies

𝑖=0

D (𝜉) = 𝐼𝑠 − 𝐷 (𝜉) 𝐷∗ (𝜉) .

𝑆 (𝜉) = (𝛽1 𝛽2 ⋅ ⋅ ⋅ 𝛽𝑠 ) ,

𝑛=0

𝑛

󵄨2 󵄨 ∑󵄨󵄨󵄨󵄨𝑓𝑖0 (𝑢)󵄨󵄨󵄨󵄨 = 1,

are the eigenvalues and eigenvectors of D(𝜉), respectively, where Ω1 , Ω2 , . . . , Ω𝑠 ensure that 𝛽1 , 𝛽2 , . . . , 𝛽𝑠 are unit vector. Let

󵄨 󵄨2 ∑ 󵄨󵄨󵄨󵄨𝑚0 (𝜉 + 2𝐴−1 𝜇𝑛 𝜋)󵄨󵄨󵄨󵄨 ≤ 1,

If there exists 𝑓𝑠0 (𝑢), . . . , 𝑓𝑛0 (𝑢), 𝑛 ≥ 𝑠, such that

𝛽1 =

𝑇

𝑠−1

󵄨 󵄨2 𝜆 𝑠 = 1 − ∑ 󵄨󵄨󵄨󵄨𝑚0 (𝜉 + 2𝐴−𝑇 𝜇𝑛 𝜋)󵄨󵄨󵄨󵄨 . 𝑛=0

𝑇 1 (𝑚∗ (𝜉 + 2𝐴−𝑇 𝜇1 𝜋), −𝑚0∗ (𝜉 + 2𝐴−𝑇 𝜇0 𝜋), 0, . . . , 0) Ω1 0

∗

∗

(64)

∗

= 𝑆 (𝜉) Λ (𝜉) 𝑆 (𝜉) = 𝐼𝑠 − 𝐷 (𝜉) 𝐷 (𝜉) ; thus, symbol functions satisfy (17). Theorems 8 and 10 give different methods to construct minimum-energy bivariate wavelet frame.

Journal of Applied Mathematics

7

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

Then we can derive the decomposition and reconstruction formulas that are similar to those of orthonormal wavelets.

Suppose that the bivariate scaling function 𝜙(𝑥) has an associated minimum-energy bivariate wavelet frame {𝜓1 , . . . , 𝜓𝑁}. Let the projection operators P𝑗 of 𝐿2 (R2 ) onto the nested subspace 𝑉𝑗 be defined by

(1) Decomposition Algorithm. Suppose that coefficients {𝑐𝑗+1,𝑙 : 𝑙 ∈ Z2 } are known. By the two-scale relations (8) and (13), we have 1 𝜙𝑗,𝑙 (𝑥) = 𝜙 ∑𝑝 (𝑥) , √𝑠 𝑘∈Z2 𝑘−𝐴𝑙 𝑗+1,𝑘 (72) 1 𝑖 𝑖 𝑖 𝜓𝑗,𝑙 (𝑥) = 𝜓 ∑𝑞 (𝑥) , 𝑖 = 1, . . . , 𝑁. √𝑠 𝑘∈Z2 𝑘−𝐴𝑙 𝑗+1,𝑘

P𝑗 𝑓 := ∑ ⟨𝑓, 𝜙𝑗,𝑘 ⟩ 𝜙𝑗,𝑘 . 𝑘∈Z2

(65)

Then the formula (12) can be rewritten as

Then, decomposition algorithm is given as

𝑁

P𝑗+1 𝑓 − P𝑗 𝑓 := ∑ ∑

𝑖=1 𝑘∈Z2

𝑖 𝑖 ⟨𝑓, 𝜓𝑗,𝑘 ⟩ 𝜓𝑗,𝑘 .

(66)

𝑐𝑗,𝑙 =

In other words, the error term 𝑔𝑗 = P𝑗+1 𝑓 − P𝑗 𝑓 between consecutive projections is given by the frame expansion:

𝑖 𝑑𝑗,𝑙 =

𝑁

𝑖 𝑖 ⟩ 𝜓𝑗,𝑘 . 𝑔𝑗 = ∑ ∑ ⟨𝑓, 𝜓𝑗,𝑘

(67)

𝑖=1 𝑘∈Z2

Suppose that the error term 𝑔𝑗 has another expansion in terms of the frames {𝜓1 , . . . , 𝜓𝑁}, that is, (68)

𝑖=1 𝑘∈Z2

𝜙𝑗+1,𝑙 (𝑥) =

𝑖=1 𝑘∈Z2

𝑁 1 ∗ 𝑖∗ 𝑖 𝜙𝑗,𝑘 (𝑥) + ∑𝑞𝑙−𝐴𝑘 𝜓𝑗,𝑘 ∑ {𝑝𝑙−𝐴𝑘 (𝑥)} . √𝑠 𝑘∈Z2 𝑖=1 (74)

𝑐𝑗+1,𝑙 =

𝑖=1 𝑘∈Z2

(73)

𝑖 = 1, . . . , 𝑁.

Take the inner products on both sides of (74) with 𝑓, we get

Then by using both (67) and (68), we have 𝑁 󵄨 󵄨2 𝑁 𝑖 𝑖 ⟩󵄨󵄨󵄨󵄨 = ∑ ∑ 𝑐𝑗,𝑘 ⟨𝑓, 𝜓𝑗,𝑘 ⟩ , (69) ⟨𝑔𝑗 , 𝑓⟩ = ∑ ∑ 󵄨󵄨󵄨󵄨⟨𝑓, 𝜓𝑗,𝑘

1 ∑ 𝑞𝑖 𝑑𝑖 , √𝑠 𝑘∈Z2 𝑘−𝐴𝑙 𝑗+1,𝑘

(2) Reconstruction Algorithm. From (26), it follows that

𝑁

𝑖 . 𝑔𝑗 = ∑ ∑ 𝑐𝑗,𝑘 𝜓𝑗,𝑘

1 𝑐 , ∑𝑝 √𝑠 𝑘∈Z2 𝑘−𝐴𝑙 𝑗+1,𝑘

𝑁 1 ∗ 𝑖∗ 𝑖 𝑐𝑗,𝑘 + ∑𝑞𝑙−𝐴𝑘 𝑑𝑗,𝑘 }. ∑ {𝑝𝑙−𝐴𝑘 √𝑠 𝑘∈Z2 𝑖=1

(75)

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

and this derives 𝑁 󵄨 󵄨2 𝑖 0 ≤ ∑ ∑ 󵄨󵄨󵄨󵄨𝑐𝑗,𝑘 − ⟨𝑓, 𝜓𝑗,𝑘 ⟩󵄨󵄨󵄨󵄨 2

Example 1. Let 𝐴 = 2𝐼2 ; then 𝑠 = 4 and

𝑖=1 𝑘∈Z

𝑁 𝑁 󵄨 󵄨2 𝑖 = ∑ ∑ 󵄨󵄨󵄨󵄨𝑐𝑗,𝑘 󵄨󵄨󵄨󵄨 − 2∑ ∑ 𝑐𝑗,𝑘 ⟨𝑓, 𝜓𝑗,𝑘 ⟩ 2 2 𝑖=1 𝑘∈Z

𝑖=1 𝑘∈Z

(70)

𝑁

󵄨 󵄨2 𝑖 + ∑ ∑ 󵄨󵄨󵄨󵄨⟨𝑓, 𝜓𝑗,𝑘 ⟩󵄨󵄨󵄨󵄨 2

𝑖=1 𝑘∈Z

This inequality means that the coefficients of the error term 𝑔𝑗 in (67) have minimal 𝑙2 -norm among all sequences {𝑐𝑗,𝑘 } which satisfy (68). We next discuss minimum-energy bivariate wavelet frames decomposition and reconstruction. For any 𝑓 ∈ 𝐿2 (R2 ), we define the coefficients 𝑐𝑗,𝑘 := ⟨𝑓, 𝜙𝑗,𝑘 ⟩ ,

𝑖 𝑑𝑗,𝑘 := ⟨𝑓, 𝜓𝑗,𝑘 ⟩

𝑖 = 1, . . . , 𝑁. (71)

𝜇2 = (0, 1)𝑇 ,

𝜇3 = (1, 1)𝑇 .

1 − 𝑒−𝑖𝜉1 1 − 𝑒−𝑖𝜉2 1 − 𝑒−𝑖(𝜉1 +𝜉2 ) 𝜙̂ (𝜉) = ⋅ ⋅ . 𝑖𝜉1 𝑖𝜉2 𝑖 (𝜉1 + 𝜉2 )

𝑁

𝑖=1 𝑘∈Z

𝜇1 = (1, 0)𝑇 ,

(76)

̂ of scaling funcSuppose that the Fourier transform 𝜙(𝜉) tion 𝜙(𝜉) is

𝑖=1 𝑘∈Z

󵄨 󵄨2 𝑁 󵄨 󵄨2 𝑖 = ∑ ∑ 󵄨󵄨󵄨󵄨𝑐𝑗,𝑘 󵄨󵄨󵄨󵄨 − ∑ ∑ 󵄨󵄨󵄨󵄨⟨𝑓, 𝜓𝑗,𝑘 ⟩󵄨󵄨󵄨󵄨 . 2 2

𝜇0 = (0, 0)𝑇 ,

(77)

Then its symbol function 𝑚0 (𝜉) satisfies 𝑚0 (𝜉) =

1 (1 + 𝑒−𝑖𝜉1 ) (1 + 𝑒−𝑖𝜉2 ) (1 + 𝑒−𝑖(𝜉1 +𝜉2 ) ) . 8

(78)

Thus, 𝑓00 (𝑢) =

𝑇 1 (1 + 𝑢(1,1) ) , 4

𝑓20 (𝑢) =

𝑓10 (𝑢) =

𝑇 1 (1 + 𝑢(1,0) ) , 4

𝑇 1 (1 + 𝑢(0,1) ) , 4

2 𝑓30 (𝑢) = . 4

(79)

8

Journal of Applied Mathematics 1

Let 𝑓40

𝑇 1 (𝑢) = (1 − 𝑢(1,0) ) , 4

𝑓50

𝑇 1 (𝑢) = (1 − 𝑢(0,1) ) , 4

𝑇 1 (1 − 𝑢(1,1) ) . 4

𝑓60 (𝑢) =

(80)

Then we have ∑6𝑙=0 |𝑓𝑖0 (𝑢)|2 = 1. By Corollary 9, there exists a minimum-energy bivariate wavelet frame associated with 𝜙(𝜉). Using Theorem 8, let √2 2 ( ( ( 𝐻1 = ( ( (

1

√2 − 2 1

1

1

√2 ( 2 1

1

( ( ( 𝐻2 = ( ( (

1

( ( ( 𝐻3 = ( ( (

) ) ) ), ) ) √2 2 )

(

√2 2

√2 2

− 1

1

1

√2 2

√2 2

) ) ) ), ) ) 1)

𝐻4 √2 √2 √2 √2 √2 2 0 0 2 −2√2 2 0 √ √ 2 2 −2 2 0 0 0 0 1( ( = ( 0 0 0 0 0 −2√2 4( 0 0 2√2 0 −2√2 0 2 2 0 0 0 −2 ( √2 √2 −√2 −2 −√2 √2

√2 0 0 ) ) 2√2) , ) 0 −2 √2 )

𝐷1 = diag (𝑢(−1,−1) , 1, 1, 1, 1, 1, 1) , 𝐷2 = diag (1, 1, 𝑢(−1,0) , 1, 1, 1, 1) ,

√2 2 √2 2

− 1

√2 2

√2 2

(

𝐷3 = diag (1, 𝑢(0,−1) , 1, 1, 1, 1, 1) .

) ) ) ), ) ) 1

1)

0 1 + 𝑢(1,1) (0,1) 1+𝑢 0 1 + 𝑢(1,0) √2+ √2𝑢(1,0) ( 1( 𝐻1∗ 𝐷1∗ 𝐻2∗ 𝐷2∗ 𝐻3∗ 𝐷3∗ 𝐻4∗ = ( 2 −2√2 4( (1,0) √ 2 − √2𝑢(1,0) 1−𝑢 (0,1) 1−𝑢 0 (1,1) 1 − 𝑢 0 (

Consequently, we obtain symbol functions 𝑚1 (𝜉) =

1 −𝑖𝜉2 √ (𝑒 ( 2 + √2𝑒−𝑖2𝜉1 ) − 2√2𝑒−𝑖(𝜉1 +𝜉2 ) ) , 8

𝑚2 (𝜉) =

1 (2𝑒−𝑖2(𝜉1 +𝜉2 ) − 2𝑒−𝑖𝜉1 𝑒−𝑖2𝜉1 ) , 8

𝑚3 (𝜉) =

1 (2 − 2𝑒−𝑖𝜉1 ) , 8

1 𝑚4 (𝜉) = 𝑒−𝑖𝜉2 (−2 + 2𝑒−𝑖2𝜉1 ) , 8 𝑚5 (𝜉) =

1 √ (− 2 + √2𝑒−𝑖(2𝜉1 +2𝜉2 ) + 𝑒−𝑖𝜉1 (−√2 + √2𝑒−𝑖2𝜉2 )) , 8

𝑚6 (𝜉) =

1 (1 + 𝑒−𝑖(2𝜉1 +2𝜉2 ) + 𝑒−𝑖𝜉1 (1 + 𝑒−𝑖2𝜉2 ) 8

(81)

Thus, we have

2𝑢(1,1) −2𝑢(0,1) 0 0 0 2𝑢(0,1) −2𝑢(1,1)

2 0 −2 0 0 −2+2𝑢(1,0) 0 0 0 −2 − 2𝑢(1,0) −2 0 2 0

−√2 + √2𝑢(1,1) −√2 + √2𝑢(0,1) 0 0 0 −√2 − √2𝑢(0,1) −√2 − √2𝑢(1,1)

1 + 𝑢(1,1) 1 + 𝑢(0,1) −1 − 𝑢(1,0) ) ) −2 ) . ) −1 + 𝑢(1,0) 1 − 𝑢(0,1) 1 − 𝑢(1,1) )

(82)

−𝑒−𝑖𝜉2 (1 + 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)

Journal of Applied Mathematics

9

̂ of scaling function Suppose that the Fourier transform 𝜙(𝜉) 𝜙(𝜉) is 1 − 𝑒−𝑖𝜉1 1 − 𝑒−𝑖𝜉2 1 − 𝑒−𝑖(𝜉1 +𝜉2 ) ⋅ ⋅ . 𝜙̂ (𝜉) = 𝑖𝜉1 𝑖𝜉2 𝑖 (𝜉1 + 𝜉2 )

(85)

𝑇

𝑇

Let 𝑧 = 𝑒−𝑖𝜉 = (𝑒−𝑖𝜉1 𝑒−𝑖𝜉2 ) , and we have 𝑧𝜇0 = 1, 𝑧𝜇1 = 𝑒−𝑖𝜉1 , 𝑢 = 𝑧𝐴 = (𝑒−𝑖(𝜉1 +𝜉2 ) , 𝑒−𝑖(𝜉1 −𝜉2 ) ). Therefore, 𝑇 1 (1+𝑢(1,0) ) , 2√2

𝑓10 (𝑢) =

𝑇 1 (1 − 𝑢(1,0) ) , 2√2

𝑓30 (𝑢) =

𝑓00 (𝑢) =

Then its symbol function 𝑚0 (𝜉) satisfies 𝑚0 (𝜉) =

1 (1 + 𝑒−𝑖𝜉1 + 𝑒−𝑖2𝜉1 ) (1 + 𝑒−𝑖𝜉2 + 𝑒−𝑖2𝜉2 ) 27

Let (86)

× (1 + 𝑒−𝑖(𝜉1 +𝜉2 ) + 𝑒−𝑖2(𝜉1 +𝜉2 ) ) . Thus,

𝑇 𝑇 1 (1 + 𝑢(0,1) + 𝑢(1,1) ) , 9

𝑓20 (𝑢) =

1 (1 + 2𝑢(0,1) ) , 9

𝑓30 (𝑢) =

𝑇 𝑇 1 (1 + 𝑢(1,0) + 𝑢(1,1) ) , 9

𝑓40 (𝑢) =

𝑇 1 (1 + 2𝑢(1,0) ) , 9

𝑓60 (𝑢) =

𝑇 1 (2 + 𝑢(0,1) ) , 9

𝑓50 (𝑢) =

𝑇 1 (2 + 𝑢(1,1) ) , 9

𝑇 1 (2 + 𝑢(1,0) ) , 9

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

3 𝑓80 (𝑢) = . 9

(87)

Let 𝑓90 (𝑢) =

𝑇 1 (1 − 𝑢(0,−1) ) , 2√2 (93)

6. Conclusions

𝑇

𝑓70 (𝑢) =

𝑓20 (𝑢) =

and we have ∑3𝑙=0 |𝑓𝑙0 (𝑢)|2 = 1. By Corollary 9, there exists a minimum-energy bivariate wavelet frame associated with 𝜙(𝜉).

𝑇 1 𝑓00 (𝑢) = (1 + 𝑢(1,1) ) , 9

𝑓10 (𝑢) =

𝑇 1 (1+𝑢(0,−1) ) . 2√2 (92)

𝑇 √6 (1 − 𝑢(1,0) ) , 9

0 𝑓11 (𝑢) =

0 𝑓10 (𝑢) =

𝑇 √6 (1 − 𝑢(0,1) ) , 9

𝑇 √6 (1 − 𝑢(1,1) ) . 9

(88)

0 2 Then we have ∑11 𝑙=0 |𝑓𝑙 (𝑢)| = 1. By Corollary 9, there exists a minimum-energy bivariate wavelet frame associated with 𝜙(𝜉). 1 ); then 𝑠 = 2 and Example 3. Let 𝐴 = ( 11 −1

𝜇0 = (0, 0)𝑇 ,

𝜇1 = (1, 0)𝑇 .

(89)

̂ of scaling funcSuppose that the Fourier transform 𝜙(𝜉) tion 𝜙(𝜉) satisfies 1 − 𝑒−𝑖𝜉1 1 − 𝑒−𝑖𝜉2 1 − 𝑒−𝑖(𝜉1 +𝜉2 ) 1 − 𝑒−𝑖(𝜉1 −𝜉2 ) ⋅ ⋅ ⋅ . 𝜙̂ (𝜉) = 𝑖𝜉1 𝑖𝜉2 𝑖 (𝜉1 + 𝜉2 ) 𝑖 (𝜉1 − 𝜉2 ) (90) Then its symbol function 𝑚0 (𝜉) satisfies 𝑚0 (𝜉) =

1 (1 + 𝑒−𝑖𝜉1 ) (1 + 𝑒−𝑖𝜉2 ) . 4

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.

(91)

References [1] R. J. Duffin and A. C. Schaeffer, “A class of nonharmonic Fourier series,” Transactions of the American Mathematical Society, vol. 72, pp. 341–366, 1952. [2] I. Daubechies, A. Grossmann, and Y. Meyer, “Painless nonorthogonal expansions,” Journal of Mathematical Physics, vol. 27, no. 5, pp. 1271–1283, 1986. [3] J. J. Benedetto and S. Li, “The theory of multiresolution analysis frames and applications to filter banks,” Applied and Computational Harmonic Analysis, vol. 5, no. 4, pp. 389–427, 1998. [4] C. K. Chui and X. L. Shi, “Orthonormal wavelets and tight frames with arbitrary real dilations,” Applied and Computational Harmonic Analysis, vol. 9, no. 3, pp. 243–264, 2000. [5] C. K. Chui and W. He, “Compactly supported tight frames associated with refinable functions,” Applied and Computational Harmonic Analysis, vol. 8, no. 3, pp. 293–319, 2000. [6] A. Petukhov, “Symmetric framelets,” Constructive Approximation, vol. 19, no. 2, pp. 309–328, 2003.

10 [7] Y. Huang and Z. Cheng, “Minimum-energy frames associated with refinable function of arbitrary integer dilation factor,” Chaos, Solitons & Fractals, vol. 32, no. 2, pp. 503–515, 2007. [8] X. Gao and C. H. Cao, “Minimum-energy wavelet frame on the interval,” Science in China F, vol. 51, no. 10, pp. 1547–1562, 2008. [9] Q. Liang and P. Zhao, “Minimum-energy multi-wavelets tight frames associatedWith two scaling functions,” in Proceedings of the International Conference on Wavelet Analysis and Pattern Recognition, vol. 1, pp. 97–101, 2010. [10] Y. D. Huang, Q. F. Li, and M. Li, “Minimum-energy multiwavelet frames on the interval with dilation factor,” Mathematical Problem in Engineering, vol. 2012, Article ID 640789, 37 pages, 2012. [11] Y. D. Huang and Q. F. Li, “Minimum-energy wavelet frame on the interval with arbitrary dilation factor,” Science China Information Science, vol. 43, no. 4, pp. 469–487, 2013. [12] M. J. Lai and J. St¨ockler, “Construction of multivariate compactly supported tight wavelet frames,” Applied and Computational Harmonic Analysis, vol. 21, no. 3, pp. 324–348, 2006. [13] J. F. Cheng and D. F. Li, “Characterization of MRA 𝐸-tight frame wavelet,” Acta Mathematica Sinica (Chinese Series), vol. 51, no. 5, pp. 877–888, 2008. [14] Q. F. Lian and Y. Z. Li, “A class of higher-dimensional FMRA wavelet frames,” Acta Mathematica Sinica (Chinese Series), vol. 52, no. 5, pp. 853–860, 2009. [15] F. Y. Zhou and Y. Z. Li, “Multivariate FMRAs and FMRA frame wavelets for reducing subspaces of 𝐿2 (R𝑑 ),” Kyoto Journal of Mathematics, vol. 50, no. 1, pp. 83–99, 2010. [16] Y. D. Huang and N. Sun, “Characterizations of 𝐴-Parseval frame wavelet,” Acta Mathematica Sinica (Chinese Series), vol. 54, no. 5, pp. 767–790, 2011. [17] Y. Zhang and T. Q. Lv, “Study of tight bivariate wavelet frames with multiscale and application in information science,” Advances in Intelligent and Soft Computing, vol. 159, pp. 125–131, 2012. [18] O. Christensen, An Introduction to Frames and Riesz Bases, Applied and Numerical Harmonic Analysis, Birkh¨aauser, Boston, Mass, USA, 2003.

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