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; nxhyd74@126.com
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.
2
Journal of Applied Mathematics
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)
Journal of Applied Mathematics
3
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)
4
Journal of Applied Mathematics
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)
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