1. Introduction
Wavelet analysis has become a common tool for data compression, feature extraction and denoising [
1,
2]. An orthonormal basis for
, which is generated by dyadic dilations and integer translates of one or several functions, is called a wavelet basis for
[
3,
4]. The decomposition of data by using wavelet bases can reveal the increment in information from a coarser approximation to a higher resolution approximation. Inspired by it, Mallat [
1] proposed the concept of multiresolution analysis.
Let be a sequence of subspaces of such that
- (i)
;
- (ii)
if and only if ;
- (iii)
there exists a such that is an orthonormal basis for .
Then is called a multiresolution analysis (MRA), and is called a scaling function.
The Meyer wavelet is among the first wavelets in the history of wavelet theory [
3]. Its construction is based on the
Meyer scaling function whose Fourier transform is
where
The corresponding
Meyer wavelet is defined by its Fourier transform:
The dyadic dilations and integer translates of the Meyer wavelet can form a wavelet basis for
. In the high dimension, the tensor product of the Meyer scaling function and Meyer wavelet can generate a
separable Meyer wavelet basis for
[
5].
Frames are an overcomplete version of bases [
5,
6]. Let
be a sequence in
. If there exists
such that
where
and
are the inner product and the norm, respectively, then
is called a
frame for
with bounds
A and
B [
1,
6].
Wavelet frames are a generalization of both wavelet bases and frames [
6]. Let
be such that their dyadic dilations and integer translates
consists of a frame for
. Then such a frame is called a
wavelet frame, and the family
is called a
wavelet frame generator [
6,
7,
8]. Compared with wavelet bases, wavelet frames can provide redundant representations of data, leading to better performances in time-frequency analysis, feature extraction, data compression and compressed sensing [
7].
Until now, the known Daubechies wavelet bases have been generalized to compactly supported (Daubechies-like) wavelet frames [
9,
10,
11], while the known bandlimited Meyer wavelet bases have not been generalized to date. In this study, we will generalize known Meyer wavelet bases into non-separable Meyer-like wavelet frames.
2. Construction of Wavelet Frames
In order to construct wavelet frames, the classic MRA is extended to the
frame MRA [
8,
12] when the condition (iii) in MRA is replaced by
where
is the characteristic function on
Q. In this case,
is called a
frame MRA for
,
is called a
frame scaling function, and
Q is called the
spectrum of frame scaling function. When
, the frame scaling function in frame MRA is just a scaling function in MRA [
8,
12].
By the bi-scale equation of frame MRA, there exists a
-periodic function
such that
is called
the filter of frame MRA [
8,
12]. Conversely, we have
Proposition 1 ([
6,
13,
14]).
If a function φ satisfies- (i)
and is continuous at ;
- (ii)
;
- (iii)
there exists a -periodic function such that ,
then φ is a frame scaling function for .
The following unitary extension principle [
9] can be used to construct wavelet frames from frame MRAs.
Proposition 2 ([
9]).
For a given frame MRA with filter and spectrum Q. Let be -periodic bounded functions and define the matrixwhere the set consists of the vertices of the cube . Let be such thatIf , then is a wavelet frame generator for .
3. One-Dimensional Meyer-like Wavelet Frames
For
, we take a point set
E satisfying
and then define a point set
F as follows (
Figure 1)
Let
be a function whose Fourier transform is
where
Then we have
Theorem 1. For , is a frame scaling function in .
Proof. For
, By Equations (1) and (2), we have
When
, we have
and
. Again by Equation (
3), we obtain
For
, we have
. Again, by
, we deduce
Again, by Equation (
1), when
, we have
. Furthermore
For
, we have
and
. Noticing that
, we can deduce that
Since the length of the interval
is
and
is
-periodic, by Equations (4) and (5), we have
where
is the characteristic function of
and
. Furthermore,
Define
and then we extend
into a
-periodic function on
.
When
, we have
and
, and so
. From this, we know that when
, Equation (
9) holds.
When
, we have
, and so
. Moreover, by Equation (
7) and
, it follows that
. Therefore, when
, Equation (
9) holds.
When
, we have
. By Equation (
3) and noticing that
,
, we obtain
for
. Therefore, when
, Equation (
9) holds.
When
, we have
. From this and Equation (
1),
. Therefore, when
, Equation (
9) holds.
By Equation (
9), we deduce that
where
is stated as in Equation (
8)
By Equations (6) and (10), noticing that and is continuous at , using Proposition 1, we know that is a frame scaling function in . □
Below we begin to construct wavelet frame generators associated with the frame scaling function
. By Equation (
8), it follows that
By Equation (
6), it follows that
and then
Therefore, the matrix
satisfies
It is clear that
M is a
-periodic point set, i.e.,
. We divide
M into two
-periodic point sets as follows
By Equations (11), (13) and (14), we have
Define
,
and
as
Now we compute
:
When
and
, we have
. By Equation (
7),
we deduce that
When
and
, we have
. Again by Equation (
13), we deduce that
When
and
, we have
and
. From this and Equations (1) and (17), we obtain
When
and
, we have
and
. From this and Equations (1) and (17), we obtain
Summarizing the above results, we have
Similarly, by Equations (6) and (16), for
, we have
By Equations (12), (14) and (15), the matrix
satisfies
Again by Equation (
16) and Proposition 2, we have
Theorem 2. For , the system is a wavelet frame generator in , where , , and are stated in Equations (18)–(20).
Remark 1. Since is just the known Meyer wavelet masked by some characteristic function, and and are just the Meyer scaling function masked by some characteristic function, the system is called a Meyer-like wavelet frame generator.
4. Non-Separable Meyer-like Frame Scaling Functions
Let
be a real-valued function satisfying the conditions:
where
b is a constant. In this section, we always take
and
. Based on
as stated in (1), we define a bivariate function
such that its Fourier transform satisfies
Define two curvilinear quadrangles
G and
as
and define
.
Theorem 3. (i) is a non-separable frame scaling function in
(ii) is circularly symmetric and .
Remark 2. In Theorem 3, by inserting a real-valued function into the width parameter of the one-dimensional Meyer-like frame scaling function, we give a novel approach to construct non-separable Meyer-like frame scaling functions in the two-dimensional case. At present, only separable frame scaling functions are circularly symmetric, the Meyer-like frame scaling function in Theorem 3 is the first non-separable frame scaling function with circular symmetry.
Proof. By Equation (
22), it is clear that
is non-separable and circularly symmetric (i.e.,
). Since
, it follows that
.
Since and is periodic, we only need to prove that .
Since
is a region bounded by four curves in the
-plane:
we can divide
into eight domains (
Figure 2):
For
, we have
(a) Let
. Then
and
, and so
Similarly, we have
. By
, we have
. Again by Equation (
7), we have
(b) Let .
For
, when
, we have
. This implies that
Since
, we have
and
, and then
. Hence,
From
, we know that
. By (3.5), we know that
Similarly, we have .
(c) Let .
For
, we have
. Furthermore,
From
, we have
. Again by Equation (
21), we have
. Similarly, we have
. Therefore,
By Equation (
5), we have
. Similarly, we have
.
Combining (a)–(c) and Equation (
24), we obtain
. Since
and
is
periodic, it follows that
By Equation (
7), we have
for
. Furthermore,
where
By
, we have
. Noticing that
by Equation (
27) and
, we have
Since
, we have
From this and Equation (
26), noticing that
and
is continuous at
, by Proposition 1, we that
is a frame scaling function of
. □
5. Non-Separable Meyer-like Wavelet Frames
In this section, we construct the wavelet frame corresponding to the frame scaling function
. Define
By Equations (28) and (29), we have
Adding the above four formulas together, we have
Define
. By Theorem 3(ii) and Equation (
26), we have
where
is the characteristic function of
. Again by Equation (
32), we have
Define a matrix
where
. By Equation (
31), we have
It means that the inner product of first and second columns of the matrix
are orthogonal. Similarly, all columns of
are mutually orthogonal. Again by Equation (
33), we have
where
It is clear that
K is a
-periodic point set. We divide
K into four
-periodic point sets as follows
Define a matrix
where
. Therefore, by Equations (34)–(36), we have
For
, we define
as
By Proposition 2, we have the following theorem:
Theorem 4. The system is a non-separable wavelet frame generator for .
Finally, we give the representation of the frame generator in Theorem 4. By Equation (29), we have Again, noticing that , we By Equation (37), we havewhereand . Example 1. Let be the cubic spline: It is well-known that and . Let Then γ satisfy Equation (21) and . Now we takeand By Equation (1), we define a bivariate function such that its Fourier transform satisfies By Theorem 3, we know that φ is a non-separable frame scaling function. Again by Equation (37) and Theorem 4, it is clear that the system is a non-separable wavelet frame generator for . 6. Conclusions
In this study, we generalized the known Meyer wavelet basis into non-separable Meyer-like wavelet frames. By using a characteristic function to mask the Fourier transform of the one-dimensional Meyer scaling function with width parameter, we can produce a family of Meyer-like frame scaling functions. Compared with the Meyer scaling function, the Meyer-like frame scaling function’s spectrum is not . After that, we construct the corresponding Meyer-like wavelet frame generator, which consists of three functions: The Fourier transform of the first function is just that of the known Meyer wavelet masked by some characteristic function; the Fourier transform of the second and third functions are just that of the Meyer scaling function masked by some characteristic function.
Separable Meyer-like wavelet frames can be constructed directly through the tensor product of one-dimensional Meyer-like wavelet frames. For the non-separable case, by inserting a real-valued function into the width parameter of the one-dimensional Meyer-like frame scaling function, we propose a novel approach to construct non-separable Meyer-like frame scaling functions. At present, only separable frame scaling functions are circularly symmetric, and our Meyer-like frame scaling function is the first non-separable frame scaling function with circular symmetry. Moreover, we construct the corresponding non-separable Meyer-like wavelet frame generator, which consists of seven functions.