# Minimum-Energy Bivariate Wavelet Frame with Arbitrary Dilation Matrix

Authors: Fengjuan Zhu; Qiufu Li; Yongdong Huang
Journal: Journal of Applied Mathematics
 (2013)

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## 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.


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## Body

## 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 Section2, 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]).
LetA 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, letℤ, 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)
LetL2(ℝ2)={f:∫ℝ2‍|f(x)|2dx<∞} and l2(ℤ2)={s:∑m∈ℤ2‍|sm|2dx<∞}. For any f,g∈L2(ℝ2), the inner product, norm, and the Fourier transform are defined, respectively, by
(1)〈f(x),g(x)〉=∫ℝ2‍f(x)g(x)-dx,∥f∥2=〈f,f〉,f^(ξ)=12π∫ℝ2‍f(x)e-ix·ξdx.(3)
For any functionf∈L2(ℝ2), fj,k(x) is defined by
(2)fj,k(x)=sj/2f(Ajx-k),j∈ℤ,k∈ℤ2.(4)
We sort the elements ofℤ2 by lexicographical order. That is, for any k=(k1,k2)T,l=(l1,l2)T,k,l∈ℤ2, and let
(3)m=(k1-l1,k2-l2);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 anyz=(z1,z2)T,z1≠0,z2≠0,k=(k1,k2)T∈ℝ2, z-1,zk are defined, respectively, by
(4)z-1=(z1-1,z2-1)T,zk=∏i=12ziki.(6)
For anyA∈ℝ2×2, zA is defined by
(5)zA=(zA1,zA2)T,where Ai is the i-th column of A.
Now, we give the definition of frame.Definition 2 (see [18]).
Letℍ 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,
(6)A∥f∥2≤∑k∈ℤ‍|〈f,fk〉|2≤B∥f∥2.
The numbers A,B are called frame bounds.
A frame is tight if we can chooseA=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
(7)f=∑k∈ℤ‍〈f,fk〉fk,∀f∈ℍ.


### 2.2. Minimum-Energy Bivariate Wavelet Frame

Definition 3 (see [18]).
A bivariate frame multiresolution analysis (FMRA) forL2(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∈ℤVj={0},⋃j∈ℤVj-=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 functionϕ(x) is called scaling function of FMRA.
SinceV0⊂V1, ϕ(x) satisfies two-scale equation (refinable equation)
(8)ϕ(x)=∑k∈ℤ2‍pkϕ(Ax-k),
where {pk}k∈ℤ2∈l2(ℤ2). The Fourier transform of (8) is
(9)ϕ^(ξ)=m0(A-Tξ)ϕ^(A-Tξ),ξ∈ℝ2,
where
(10)m0(ξ)=1|det(A)|∑k∈ℤ2‍pke-iξTk=1s∑k∈ℤ2‍pkzk,
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.
Letϕ∈L2(ℝ2) satisfies ϕ^∈L∞ and ϕ^ 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):
(11)∑k∈ℤ2‍|〈f,ϕ1,k〉|2=∑k∈ℤ2‍|〈f,ϕ0,k〉|2+∑l=1N‍∑k∈ℤ2‍|〈f,ψ0,kl〉|2.
By the Parseval identity, minimum-energy bivariate wavelet frameΨ 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
(12)∑k∈ℤ2‍〈f,ϕ1,k〉ϕ1,k=∑k∈ℤ2‍〈f,ϕ0,k〉ϕ0,k+∑l=1N‍∑k∈ℤ2‍〈f,ψ0,kl〉ψ0,kl,∀f∈L2(ℝ2).
The interpretation of minimum-energy bivariate wavelet frame will be clarified later.
ConsiderΨ={ψ1,ψ2,…,ψN}⊂V1, with
(13)ψl(x)=∑k∈ℤ2‍qklϕ(Ax-k),l=1,2,…,N.
Using Fourier transform on the previous equation, we can get their symbols as follows:
(14)ml(ξ)=1s∑k∈ℤ2‍qkle-iξTk=1s∑k∈ℤ2‍qklzk,l=1,2,…,N,
where z=e-iξT.
Withm0(ξ),m1(ξ),…,mN(ξ), we formulate the s×(N+1) matrix M(ξ):
(15)M(ξ)=(m0(ξ)m1(ξ)⋯mN(ξ)m0(ξ+2A-Tμ1π)m1(ξ+2A-Tμ1π)⋯mN(ξ+2A-Tμ1π)⋮⋮⋮m0(ξ+2A-Tμs-1π)m1(ξ+2A-Tμs-1π)⋯mN(ξ+2A-Tμs-1π)),
and M*(ξ) denotes the complex conjugate of the transpose of M(ξ).


## 2.1. Basic Concept and Notation

Let us recall the concept of dilation matrix and give some notations.Definition 1 (see [18]).
LetA 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, letℤ, 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)
LetL2(ℝ2)={f:∫ℝ2‍|f(x)|2dx<∞} and l2(ℤ2)={s:∑m∈ℤ2‍|sm|2dx<∞}. For any f,g∈L2(ℝ2), the inner product, norm, and the Fourier transform are defined, respectively, by
(1)〈f(x),g(x)〉=∫ℝ2‍f(x)g(x)-dx,∥f∥2=〈f,f〉,f^(ξ)=12π∫ℝ2‍f(x)e-ix·ξdx.(3)
For any functionf∈L2(ℝ2), fj,k(x) is defined by
(2)fj,k(x)=sj/2f(Ajx-k),j∈ℤ,k∈ℤ2.(4)
We sort the elements ofℤ2 by lexicographical order. That is, for any k=(k1,k2)T,l=(l1,l2)T,k,l∈ℤ2, and let
(3)m=(k1-l1,k2-l2);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 anyz=(z1,z2)T,z1≠0,z2≠0,k=(k1,k2)T∈ℝ2, z-1,zk are defined, respectively, by
(4)z-1=(z1-1,z2-1)T,zk=∏i=12ziki.(6)
For anyA∈ℝ2×2, zA is defined by
(5)zA=(zA1,zA2)T,where Ai is the i-th column of A.
Now, we give the definition of frame.Definition 2 (see [18]).
Letℍ 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,
(6)A∥f∥2≤∑k∈ℤ‍|〈f,fk〉|2≤B∥f∥2.
The numbers A,B are called frame bounds.
A frame is tight if we can chooseA=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
(7)f=∑k∈ℤ‍〈f,fk〉fk,∀f∈ℍ.


## 2.2. Minimum-Energy Bivariate Wavelet Frame

Definition 3 (see [18]).
A bivariate frame multiresolution analysis (FMRA) forL2(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∈ℤVj={0},⋃j∈ℤVj-=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 functionϕ(x) is called scaling function of FMRA.
SinceV0⊂V1, ϕ(x) satisfies two-scale equation (refinable equation)
(8)ϕ(x)=∑k∈ℤ2‍pkϕ(Ax-k),
where {pk}k∈ℤ2∈l2(ℤ2). The Fourier transform of (8) is
(9)ϕ^(ξ)=m0(A-Tξ)ϕ^(A-Tξ),ξ∈ℝ2,
where
(10)m0(ξ)=1|det(A)|∑k∈ℤ2‍pke-iξTk=1s∑k∈ℤ2‍pkzk,
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.
Letϕ∈L2(ℝ2) satisfies ϕ^∈L∞ and ϕ^ 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):
(11)∑k∈ℤ2‍|〈f,ϕ1,k〉|2=∑k∈ℤ2‍|〈f,ϕ0,k〉|2+∑l=1N‍∑k∈ℤ2‍|〈f,ψ0,kl〉|2.
By the Parseval identity, minimum-energy bivariate wavelet frameΨ 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
(12)∑k∈ℤ2‍〈f,ϕ1,k〉ϕ1,k=∑k∈ℤ2‍〈f,ϕ0,k〉ϕ0,k+∑l=1N‍∑k∈ℤ2‍〈f,ψ0,kl〉ψ0,kl,∀f∈L2(ℝ2).
The interpretation of minimum-energy bivariate wavelet frame will be clarified later.
ConsiderΨ={ψ1,ψ2,…,ψN}⊂V1, with
(13)ψl(x)=∑k∈ℤ2‍qklϕ(Ax-k),l=1,2,…,N.
Using Fourier transform on the previous equation, we can get their symbols as follows:
(14)ml(ξ)=1s∑k∈ℤ2‍qkle-iξTk=1s∑k∈ℤ2‍qklzk,l=1,2,…,N,
where z=e-iξT.
Withm0(ξ),m1(ξ),…,mN(ξ), we formulate the s×(N+1) matrix M(ξ):
(15)M(ξ)=(m0(ξ)m1(ξ)⋯mN(ξ)m0(ξ+2A-Tμ1π)m1(ξ+2A-Tμ1π)⋯mN(ξ+2A-Tμ1π)⋮⋮⋮m0(ξ+2A-Tμs-1π)m1(ξ+2A-Tμs-1π)⋯mN(ξ+2A-Tμs-1π)),
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 thatA is a dilation matrix; let
(16)𝒜=(ei2πμ1TA-1μ1ei2πμ1TA-1μ2⋯ei2πμ1TA-1μs-1ei2πμ2TA-1μ1ei2πμ2TA-1μ2⋯ei2πμ2TA-1μs-1⋮⋮⋮ei2πμs-1TA-1μ1ei2πμs-1TA-1μ2⋯ei2πμs-1TA-1μs-1);
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 symbolsml(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 ϕ^ is continuous at 0 and ϕ^(0)=1 and ϕ(x) generates a nested closed 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)
(17)M(ξ)M*(ξ)=Is,∀ξ∈ℝ2.(3)
(18)αml=∑k∈ℤ2‍(pm-Ak*pl-Ak+∑i=1N‍qm-Aki*ql-Aki)-sδml=0,∀m,l∈ℤ2.Proof.
By using the two-scale relations (8) and (13) and notation αml, then formula (12) is equivalent to
(19)∑m,l∈ℤ2‍αml〈f,ϕ(Ax-m)〉ϕ(Ax-l)=0,∀f∈L2(ℝ2).
On the other hand, formula (17) can be reformulated as
(20)∑l=0N‍|ml(ξ)|2=1,∑l=0N‍ml(ξ)ml*(ξ+2A-1μjπ)=0,j=1,…,s-1
which is equivalent to
(21)∑l=0N‍ml(ξ)∑j=0s-1‍ml*(ξ+2A-1μjπ)=1,∑l=0N‍ml(ξ)(m0*(ξ)-∑j=1s-1‍ml*(ξ+2A-1μjπ))=1,∑l=0N‍ml(ξ)(∑j=0s-1‍ml*(ξ+2A-1μjπ)-2ml(ξ+2A-1μnπ))=1,n=1,…,s-1;ξ∈ℝ2,
or
(22)∑l=0N‍ml(ξ)∑k∈ℤ2‍q-Akl*zAk=1,∑l=0N‍ml(ξ)(∑n=1s-1‍∑k∈ℤ2‍qμn-Akl*zAk-μn)=s-1,∑l=0N‍ml(ξ)(∑n=1s-1‍ei2μjTA-1μnπ∑k∈ℤ2‍qμn-Akl*zAk-μn)=-1,j=1,…,s-1;ξ∈ℝ2,
where qk0=pk,k∈ℤ2. Since 𝒜 is an invertible matrix, the previous equation is equivalent to
(23)∑l=0N‍ml(ξ)∑k∈ℤ2‍qμn-Akl*zAk-μn=1,n=0,1,…,s-1.
We multiply the identities in (23) by zμnϕ^(A-Tξ),n=0,1,…,s-1, respectively, where z=e-iξTA-1, and we get
(24)ϕ^(A-Tξ)zμn=∑l=0N‍∑k∈ℤ2‍qμn-Akl*zAkψ^l(ξ),n=0,1,…,s-1;
let ψ0=ϕ. Take the Fourier transform on the two sides of the previous formula, (23) is equivalent to
(25)sϕ(Ax-μn)=∑l=0N‍∑k∈ℤ2‍qμn-Akl*ψl(x-k),n=0,1,…,s-1.
Thus,
(26)sϕ(Ax-μ)=∑l=0N‍∑k∈ℤ2‍qμ-Akl*ψl(x-k),∀μ∈ℤ2.
By using the two-scale relations (8) and (13), we can rewrite formula (26) as
(27)∑m∈ℤ2‍αmlϕ(Ax-m)=0,∀l∈ℤ2.
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
(28)βl(f)=〈f,∑m∈ℤ2‍αmlϕ(Ax-m)〉,l∈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 nonzero. Now, since ϕ^(ξ) is a nontrivial function, by taking the Fourier transform of (19), it follows that the trigonometric polynomial ∑l∈ℤ2‍βl(f)e-iξTA-Tl≡0. Obviously, βl(f)=0,l∈ℤ2. By choosing f=∑m∈ℤ2‍αmlϕ(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.Theorem6 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 functionϕ∈L2(ℝ2), with ϕ^ continuous at 0 and ϕ^(0)=1, and m0(ξ) is the symbol function of ϕ. Let Ψ={ψ1,ψ2,…,ψN} be the minimum-energy multiwavelet frames associated with ϕ; then
(29)∑n=0s-1‍|m0(ξ+2A-1μnπ)|2≤1,∀ξ∈ℝ2.Proof.
LetD(ξ) be the first column of M(ξ) and M(ξ)=(D(ξ),Q(ξ)). Then,
(30)D(ξ)D*(ξ)+Q(ξ)Q*(ξ)=Is.
Since Q(ξ)Q*(ξ) is a Hermitian matrix, the matrix Is-D(ξ)D*(ξ) is positive semidefinite.
We have(31)(IsD(ξ)D*(ξ)1)(Is-D(ξ)-D*(ξ)1)=(Is-D(ξ)D*(ξ)1-D*(ξ)D(ξ))det(IsD(ξ)D*(ξ)1)=det(IsD(ξ)1-D*(ξ)D(ξ)),det(Is-D(ξ)-D*(ξ)1)=det(Is-D(ξ)1-D*(ξ)D(ξ)).
Therefore,
(32)det(Is-D(ξ)D*(ξ))(1-D*(ξ)D(ξ))=(1-D*(ξ)D(ξ))(1-D*(ξ)D(ξ)),
that is,
(33)1-D*(ξ)D(ξ)≥0.
The proof of Theorem7 is completed.According to the Theorem7, 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.Letz=e-iξT; then
(34)m0(ξ)=1s∑k∈ℤ2‍pke-iξTk=1s∑i=0s-1‍∑k∈ℤ2‍pμi+Ake-iξT(μi+Ak)=1s∑i=0s-1‍e-iξTμi∑k∈ℤ2‍pμi+Ake-iξTAk=1s∑i=0s-1‍zμi∑k∈ℤ2‍pμi+AkzAk=1s∑i=0s-1‍zμifi0(zA),
where fi0(x)=(1/s)∑k∈ℤ2‍pμi+Akxk,x∈ℝ2. Similarly,
(35)ml(ξ)=1s∑i=0s-1‍zμifil(zA),l=1,2,…,N,
where fil(x)=(1/s)∑k∈ℤ2‍qμi+Aklxk,x∈ℝ2. Let
(36)C(z)=1s×(zμ0e-i2πμ0TA-1μ0zμ1e-i2πμ0TA-1μ1⋯zμs-1e-i2πμ0TA-1μs-1zμ0e-i2πμ1TA-1μ0zμ1e-i2πμ1TA-1μ1⋯zμs-1e-i2πμ1TA-1μs-1⋮⋮⋮zμ0e-i2πμs-1TA-1μ0zμ1e-i2πμs-1TA-1μ1⋯zμ3e-i2πμs-1TA-1μs-1);
thus,
(37)M(ξ)=C(z)(f00(zA)f01(zA)⋯f0N(zA)f10(zA)f11(zA)⋯f1N(zA)⋮⋮⋮fs-10(zA)fs-11(zA)⋯fs-1N(zA)).
Since C(z) is a unitary matrix, condition (17) is equivalent to
(38)(f00(zA)f01(zA)⋯f0N(zA)f10(zA)f11(zA)⋯f1N(zA)⋮⋮⋮fs-10(zA)fs-11(zA)⋯fs-1N(zA))×(f00(zA)f01(zA)⋯f0N(zA)f10(zA)f11(zA)⋯f1N(zA)⋮⋮⋮fs-10(zA)fs-11(zA)⋯fs-1N(zA))*=Is.For convenience, letu=zA, condition (29) can be rewritten as ∑n=0s-1‍|fn0(u)|2≤1.If there existsfs0(u) such that
(39)∑n=0s‍|fn0(u)|2=1,
then, we have the following Theorem 8.Theorem 8.
Letϕ(x)∈L2(ℝ2) be a compactly supported refinable function, with ϕ^ continuous at 0 and ϕ^(0)=1, and its symbol function satisfies
(40)∑n=0s-1‍|fn0(u)|2≤1.
If there exists fs0(u) such that
(41)∑n=0s‍|fn0(u)|2=1,
then there exists a minimum-energy bivariate wavelet frame associated with ϕ(x).Proof.
Under the assumption, we know that the vector(42)f:=[f00(u),f10(u),…,fs-10(u),fs0(u)]T
is a unit vector.
Construct diagonal matrixD0=diag(ut0,ut1,…,uts) such that
(43)f1=D0f=[ut0f00(u),ut1f10(u),…,utsfs0(u)]T=∑j=(0,0)K‍ajuj,t0,t1,…,ts∈ℤ2,
where aj∈ℝs+1 with a0≠0 and aK≠0. It is clear that f1 is a unit vector:
(44)f1*f1=(∑j=(0,0)K‍ajuj)*(∑j=(0,0)K‍ajuj)=1,∀|u|=1,
and consequently a0TaK=0. We next consider the (s+1)×(s+1) Householder matrix:
(45)H1=Is+1-2∥v∥2vvT,
where v=aK±∥aK∥e1, with e1=(1,0,…,0)s+1T, and the + or − signs are so chosen that v≠0. Then
(46)H1aK=±∥aK∥e1.
Since Householder matrix is orthogonal matrix, we have
(47)(H1a0)T(H1aK)=a0TH1TH1aK=a0TaK=0.
By the previous equation, the first component of H1a0 is 0. Now H1f1=∑j=(0,0)K‍(H1aj)uj, we construct diagonal matrix D1=diag(ut(1),1,…,1), t(1)∈ℤ2 such that
(48)f2=D1H1f1=D1∑j=(0,0)K‍(H1aj)uj=∑j=(0,0)K‍aj(1)uj
is also a unit vector and K1<K,a0(1)≠0,aK1(1)≠0.
Similarly, we define the Householder matrix:(49)H2=Is+1-2∥v-∥2v-v-T,
where v-=aK1±∥aK1∥e1≠0, and D2=diag(ut(2),1,…,1) such that
(50)f3=D2H2f2=D2∑j=(0,0)K1‍(H2aj(1))uj=∑j=(0,0)K1‍aj(2)uj
is also a unit vector and K2<K1,a0(2)≠0,aK2(2)≠0.
Since every component off is a finite sum, we repeat this procedure finite times to get some unitary matrices DL,HL,DL-1,HL-1,…,H2,D1,H1,
(51)DLHLDL-1HL-1⋯H2D1H1D0f=e1.
That is, f is the first column of the unitary matrix
(52)H:=D0*H0*D1*H2*⋯HL-1*DL-1*HL*DL*.
Let
(53)H=(f00(u)f01(u)⋯f0s(u)f10(u)f11(u)⋯f1s(u)⋮⋮⋮fs0(u)fs1(u)⋯fss(u));
then, H satisfies (38). In the formula (15), we let
(54)ml(ξ)=1s∑i=0s-1‍zμifil(zA),l=1,…,s.
Then,we can obtain the formula (17). The proof of Theorem 8 is completed.Corollary 9.
Letϕ(x)∈L2(ℝ2) be a compactly supported refinable function, with ϕ^ continuous at 0 and ϕ^(0)=1, and its symbol function satisfies
(55)∑n=0s-1‍|m0(ξ+2A-1μnπ)|2≤1,∀ξ∈ℝ2.
If there exists fs0(u),…,fn0(u),n≥s, such that
(56)∑i=0n‍|fi0(u)|2=1,
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 thatm0 satisfies (29); let
(57)𝒟(ξ)=Is-D(ξ)D*(ξ).
Then
(58)λ1=⋯=λs-1=1,λs=1-∑n=0s-1‍|m0(ξ+2A-Tμnπ)|2.β1=1Ω1(m0*(ξ+2A-Tμ1π),-m0*(ξ+2A-Tμ0π),0,…,0)Tβk=1Ωk(∑j=0k-1‍-m0(ξ+2A-Tμ0π)m0*(ξ+2A-Tμkπ),-m0(ξ+2A-Tμ1π)m0*(ξ+2A-Tμkπ),…,-m0(ξ+2A-Tμk-1π)m0*(ξ+2A-Tμkπ),∑j=0k-1‍|m0(ξ+2A-Tμjπ)|2,0,…,0)Tk=2,3,…,s-1,βs=1Ωs(m0(ξ+2A-Tμ0π),m0(ξ+2A-Tμ1π),…,m0(ξ+2A-Tμs-1π))T
are the eigenvalues and eigenvectors of 𝒟(ξ), respectively, where Ω1,Ω2,…,Ωs ensure that β1,β2,…,βs are unit vector. Let
(59)S(ξ)=(β1β2⋯βs),Λ(ξ)=diag(1,…,1,λs),
then,
(60)Is-D(ξ)D*(ξ)=S(ξ)Λ(ξ)S*(ξ).Theorem 10.
Letϕ(x)∈L2(ℝ2) be a compactly supported refinable function, with ϕ^ continuous at 0 and ϕ^(0)=1, and its symbol function satisfies
(61)∑n=0s-1‍|m0(ξ+2A-1μnπ)|2≤1,∀ξ∈ℝ2.
If there exists fs0(u) satisfies
(62)∑i=0s‍|fi0(u)|2=1,
then the symbol function of minimum-energy bivariate wavelet frame associated with ϕ(x) is the first row of the matrix:
(63)Q(ξ)=U(ξ)Λ(ξ)S(ξ),
where U(ξ) is any s×s unitary matrix, and S(ξ),Λ(ξ) are defined in (59).Proof.
By Theorem8, 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
(64)Q(ξ)Q*(ξ)=S(ξ)Λ(ξ)U(ξ)(S(ξ)Λ(ξ)U(ξ))*=S(ξ)Λ(ξ)S*(ξ)=Is-D(ξ)D*(ξ);
thus, symbol functions satisfy (17).Theorems8 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 functionϕ(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
(65)Pjf:=∑k∈ℤ2‍〈f,ϕj,k〉ϕj,k.
Then the formula (12) can be rewritten as
(66)Pj+1f-Pjf:=∑i=1N∑k∈ℤ2〈f,ψj,ki〉ψj,ki.
In other words, the error term gj=Pj+1f-Pjf between consecutive projections is given by the frame expansion:
(67)gj=∑i=1N‍∑k∈ℤ2‍〈f,ψj,ki〉ψj,ki.Suppose that the error termgj has another expansion in terms of the frames {ψ1,…,ψN}, that is,
(68)gj=∑i=1N‍∑k∈ℤ2‍cj,kψj,ki.
Then by using both (67) and (68), we have
(69)〈gj,f〉=∑i=1N‍∑k∈ℤ2‍|〈f,ψj,ki〉|2=∑i=1N‍∑k∈ℤ2‍cj,k〈f,ψj,ki〉-,
and this derives
(70)0≤∑i=1N‍∑k∈ℤ2‍|cj,k-〈f,ψj,ki〉|2=∑i=1N‍∑k∈ℤ2‍|cj,k|2-2∑i=1N‍∑k∈ℤ2‍cj,k〈f,ψj,ki〉-+∑i=1N‍∑k∈ℤ2‍|〈f,ψj,ki〉|2=∑i=1N‍∑k∈ℤ2‍|cj,k|2-∑i=1N‍∑k∈ℤ2‍|〈f,ψj,ki〉|2.
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 anyf∈L2(ℝ2), we define the coefficients
(71)cj,k:=〈f,ϕj,k〉,dj,k:=〈f,ψj,ki〉i=1,…,N.
Then we can derive the decomposition and reconstruction formulas that are similar to those of orthonormal wavelets.(1) Decomposition Algorithm. Suppose that coefficients {cj+1,l:l∈ℤ2} are known. By the two-scale relations (8) and (13), we have
(72)ϕj,l(x)=1s∑k∈ℤ2‍pk-Alϕj+1,k(x),ψj,li(x)=1s∑k∈ℤ2‍qk-Aliψj+1,ki(x),i=1,…,N.
Then, decomposition algorithm is given as
(73)cj,l=1s∑k∈ℤ2‍pk-Alcj+1,k,dj,li=1s∑k∈ℤ2‍qk-Alidj+1,ki,i=1,…,N.(2) Reconstruction Algorithm. From (26), it follows that
(74)ϕj+1,l(x)=1s∑k∈ℤ2‍{pl-Ak*ϕj,k(x)+∑i=1N‍ql-Aki*ψj,ki(x)}.
Take the inner products on both sides of (74) with f, we get
(75)cj+1,l=1s∑k∈ℤ2‍{pl-Ak*cj,k+∑i=1N‍ql-Aki*dj,ki}.


## 5. Numerical Examples

In this section, we present some numerical examples to show the effectiveness of the proposed methods.Example 1.
LetA=2I2; then s=4 and
(76)μ0=(0,0)T,μ1=(1,0)T,μ2=(0,1)T,μ3=(1,1)T.
Suppose that the Fourier transformϕ^(ξ) of scaling function ϕ(ξ) is
(77)ϕ^(ξ)=1-e-iξ1iξ1·1-e-iξ2iξ2·1-e-i(ξ1+ξ2)i(ξ1+ξ2).
Then its symbol function m0(ξ) satisfies
(78)m0(ξ)=18(1+e-iξ1)(1+e-iξ2)(1+e-i(ξ1+ξ2)).
Thus,
(79)f00(u)=14(1+u(1,1)T),f10(u)=14(1+u(0,1)T),f20(u)=14(1+u(1,0)T),f30(u)=24.
Let
(80)f40(u)=14(1-u(1,0)T),f50(u)=14(1-u(0,1)T),f60(u)=14(1-u(1,1)T).
Then we have ∑l=06‍|fi0(u)|2=1. By Corollary 9, there exists a minimum-energy bivariate wavelet frame associated with ϕ(ξ). Using Theorem 8, let
(81)H1=(22-22111112222),H2=(1122-221222211),H3=(122-2211122221),H4=14(2222222002-2220022-220000000000-222200220-220022000-2-222-2-2-222),D1=diag(u(-1,-1),1,1,1,1,1,1),D2=diag(1,1,u(-1,0),1,1,1,1),D3=diag(1,u(0,-1),1,1,1,1,1).
Thus, we have(82)
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.Consequently, we obtain symbol functions(83)m1(ξ)=18(e-iξ2(2+2e-i2ξ1)-22e-i(ξ1+ξ2)),m2(ξ)=18(2e-i2(ξ1+ξ2)-2e-iξ1e-i2ξ1),m3(ξ)=18(2-2e-iξ1),m4(ξ)=18e-iξ2(-2+2e-i2ξ1),m5(ξ)=18(-2+2e-i(2ξ1+2ξ2)+e-iξ1(-2+2e-i2ξ2)),m6(ξ)=18(1+e-i(2ξ1+2ξ2)+e-iξ1(1+e-i2ξ2)-e-iξ2(1+e-i2ξ1)-2e-i(ξ1+ξ2)).
The functions generated by the previous symbol functions are minimum-energy bivariate wavelet frame functions associated with ϕ(x).Example 2.
LetA=3I2; then s=9 and
(84)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.
Suppose that the Fourier transform ϕ^(ξ) of scaling function ϕ(ξ) is
(85)ϕ^(ξ)=1-e-iξ1iξ1·1-e-iξ2iξ2·1-e-i(ξ1+ξ2)i(ξ1+ξ2).
Then its symbol function m0(ξ) satisfies
(86)m0(ξ)=127(1+e-iξ1+e-i2ξ1)(1+e-iξ2+e-i2ξ2)×(1+e-i(ξ1+ξ2)+e-i2(ξ1+ξ2)).
Thus,
(87)f00(u)=19(1+u(1,1)T),f10(u)=19(1+u(0,1)T+u(1,1)T),f20(u)=19(1+2u(0,1)T),f30(u)=19(1+u(1,0)T+u(1,1)T),f40(u)=19(1+2u(1,0)T),f50(u)=19(2+u(1,1)T),f60(u)=19(2+u(0,1)T),f70(u)=19(2+u(1,0)T),f80(u)=39.
Let
(88)f90(u)=69(1-u(1,0)T),f100(u)=69(1-u(0,1)T),f110(u)=69(1-u(1,1)T).
Then we have ∑l=011‍|fl0(u)|2=1. By Corollary 9, there exists a minimum-energy bivariate wavelet frame associated with ϕ(ξ).Example 3.
LetA=(111-1); then s=2 and
(89)μ0=(0,0)T,μ1=(1,0)T.
Suppose that the Fourier transformϕ^(ξ) of scaling function ϕ(ξ) satisfies
(90)ϕ^(ξ)=1-e-iξ1iξ1·1-e-iξ2iξ2·1-e-i(ξ1+ξ2)i(ξ1+ξ2)·1-e-i(ξ1-ξ2)i(ξ1-ξ2).
Then its symbol function m0(ξ) satisfies
(91)m0(ξ)=14(1+e-iξ1)(1+e-iξ2).
Let z=e-iξT=(e-iξ1e-iξ2)T, and we have zμ0=1,zμ1=e-iξ1, u=zA=(e-i(ξ1+ξ2),e-i(ξ1-ξ2)). Therefore,
(92)f00(u)=122(1+u(1,0)T),f10(u)=122(1+u(0,-1)T).
Let
(93)f20(u)=122(1-u(1,0)T),f30(u)=122(1-u(0,-1)T),
and we have ∑l=03‍|fl0(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.



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*Source: 896050-2013-07-21.xml*