Journal of Applied Mathematics

Journal of Applied Mathematics / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 272801 | 19 pages | https://doi.org/10.1155/2011/272801

Shift Unitary Transform for Constructing Two-Dimensional Wavelet Filters

Academic Editor: F. Marcellán
Received13 Mar 2011
Revised27 May 2011
Accepted08 Jun 2011
Published14 Sep 2011

Abstract

Due to the difficulty for constructing two-dimensional wavelet filters, the commonly used wavelet filters are tensor-product of one-dimensional wavelet filters. In some applications, more perfect reconstruction filters should be provided. In this paper, we introduce a transformation which is referred to as Shift Unitary Transform (SUT) of Conjugate Quadrature Filter (CQF). In terms of this transformation, we propose a parametrization method for constructing two-dimensional orthogonal wavelet filters. It is proved that tensor-product wavelet filters are only special cases of this parametrization method. To show this, we introduce the SUT of one-dimensional CQF and present a complete parametrization of one-dimensional wavelet system. As a result, more ways are provided to randomly generate two-dimensional perfect reconstruction filters.

1. Introduction

In her celebrated paper [1], Daubechies constructed a family of compactly supported orthonormal scaling functions and their corresponding orthonormal wavelets. Since then, wavelets with compact support have been found to be very useful in applications (see [2] and references therein). By now, the theory for the construction of one-dimensional wavelets is well developed [1, 39]. But, there still exists many open problems for the construction of multidimensional wavelets ([1013], etc.).

To apply wavelet methods to digital image processing, two-dimensional wavelets have to be constructed. The most common wavelets used for image processing are tensor-product of one-dimensional wavelets (separable wavelets). Nevertheless, separable wavelets have a number of drawbacks [11]. Nonseparable wavelets offer the hope of more isotropic analysis ([1416], etc.). Many efforts have been made on constructing nonseparable wavelets. However, up to now, only a few constructions have been published. Cohen and Daubechies [14] used the univariate construction [1] to produce nonseparable scaling function with higher accuracy. Continuous nonseparable scaling functions were constructed by He and Lai [12] and Kovă𝑐ević [15]. Arbitrarily smooth nonseparable orthogonal wavelets were constructed by Ayache [10] and Belogay and wang [11]. Recently, Lai [13] proposed a constructive method to find compactly supported orthonormal wavelets for any given compactly supported scaling function in the multivariate setting.

In some applications of wavelets, such as wavelet-based watermarking [17, 18], parametrization of two-dimensional wavelet filters is preferred. To make some wavelet-based watermarking schemes more robust, we need to create as many ways as possible to randomly generate perfect reconstruction filters [17]. The ample choices of wavelet filters will increase the difficulty for counterfeiters to gain the exact knowledge of the filters (see [1720]). But in methods available, to derive two-dimensional wavelet filters, one has to solve transcendental constraints for the parameters. Hence, wavelet filters used in wavelet-based watermarking schemes, such as [1720], are only tensor-product of one-dimensional wavelets.

In this paper, a transformation that we refer to as Shift Unitary Transform (SUT) of Conjugate Quadrature Filter (CQF) is proposed. In terms of this transformation, we present a parametrization method for constructing two-dimensional orthogonal wavelet filters. The choosing of the parameters is not restricted by any implicit condition. It is proved that tensor-product wavelet filters are only special cases of this parametrization method. Therefore, more ways are provided to randomly generate perfect reconstruction filters. The ample choices of wavelet filters will increase the difficulty for counterfeiters to gain the exact knowledge of the filters and make watermarking schemes based on wavelet filters more robust [1720].

First of all, it should be pointed out that all the filters along the paper are FIR (finite impulse response). For a matrix 𝐴, we denote its transpose by 𝐴𝑇 in this paper.

To show that tensor-product wavelet filters are only special cases of our construction, we introduce the SUT of one-dimensional CQF and present a complete parametrization of one-dimensional wavelet system in Section 2. In Section 3, we show that tensor-product orthogonal wavelet filters can be constructed by SUT of CQF. Then, based on SUT of two-dimensional CQF, a parameterized method is presented for constructing real-valued two-dimensional orthogonal wavelet filters. Finally, nonseparable wavelet filters are derived. Conclusion remarks are given in Section 4.

2. Parametrization of One-Dimensional Wavelet Filters

To construct one-dimensional orthogonal scaling function 𝜑(𝑥)=2𝑘𝑘𝜑(2𝑥𝑘),(2.1) we need construct sequences {𝑘}𝑘 such that (see [4, 21] and many others) 𝑘𝑘=2,(2.2)𝑘2𝑘=𝑘2𝑘+1,(2.3)𝑘𝑘𝑘+2𝛼=𝛿0,𝛼,𝛼,(2.4) where 𝛿 denotes the Kronecker delta 𝛿𝛼,𝛽=0,𝛼𝛽,1,𝛼=𝛽.𝛼,𝛽.(2.5) The sequence {𝑘}𝑘 which satisfies (2.4) is called a one-dimensional CQF, and the sequence {𝑘}𝑘, which satisfies (2.2), (2.3), (2.4) simultaneously, is called one-dimensional orthogonal low-pass wavelet filter.

Definition 2.1. Let {𝑛}𝑛 be a one-dimensional CQF and 𝐺 an arbitrary 2×2 orthogonal matrix. {𝑛}𝑛 is called the shift unitary transform (SUT) of {𝑛}𝑛by 𝐺 if {𝑛}𝑛 satisfies Υ𝑖=Λ𝑖𝐺,𝑖,(2.6) and {𝑛}𝑛 is called the inverse shift unitary transform (ISUT) of {𝑛}𝑛 by 𝐺 if {𝑛}𝑛 satisfies Λ𝑖=Υ𝑖𝐺,𝑖,(2.7) where for any 𝑖𝑍, Υ𝑖=2𝑖2𝑖+1𝑇,Λ𝑖=2𝑖2𝑖1𝑇,Λ𝑖=2𝑖2𝑖1𝑇,Υ𝑖=2𝑖2𝑖+1𝑇.(2.8)

The SUT and the ISUT of {𝑛}𝑛 by 𝐺 are, respectively, denoted by 𝑛𝑛=G𝑛𝑛,𝑛𝑛=𝐆𝑛𝑛.(2.9) By directly calculating, we have the following results.

Lemma 2.2. If {𝑛}𝑛 is a one-dimensional CQF, then {𝑛}𝑛=𝐆{𝑛}𝑛 and {𝑛}𝑛=𝐆{𝑛}𝑛 are one-dimensional CQFs.

Lemma 2.3. If {𝑛}𝑛=𝐆{𝑛}𝑛, then {𝑛}𝑛=(𝐆𝑇){𝑛}𝑛.

Therefore, for a one-dimensional CQF, different one-dimensional CQFs can be derived when we choose different orthogonal matrices 𝐺. For a sequence {𝑠𝑛}𝑛, letting Γ={𝑛𝑠𝑛0}, we call Γ the support of {𝑠𝑛}𝑛. If {𝑠𝑛}𝑛 is a one-dimensional CQF and the support of it is in {0,1}, we call {𝑠𝑛}𝑛 a simple one-dimensional CQF.

Theorem 2.4. If {𝑛}𝑛 is a one-dimensional CQF, it can be constructed from a simple one-dimensional CQF by a series of SUT.

Proof. It only needs to prove that by a series of ISUT of {𝑛}𝑛, we can get a simple one-dimensional CQF.
Without loss of generality, we assume that the support of {𝑛}𝑛 is in {0,1,2,, 2𝑁1} and 02𝑁10. We will prove this theorem by induction.
We see that the theorem is true for 𝑁=1. Assume it is true for the case 𝑁𝐿(𝐿1,𝐿). We now prove it is true for the case 𝑁=𝐿+1. Suppose that {𝑛}𝑛 is a one-dimensional CQF and 02𝐿+10. We denote that ,𝐺=cos𝜃sin𝜃sin𝜃cos𝜃𝑛𝑛=𝐆𝑛𝑛,(2.10) where tan𝜃=1/0. Recall that {𝑛}𝑛 is a one-dimensional CQF, that is, 𝑛𝑛𝑛+2𝑚=𝛿0,𝑚, which implies that 02𝐿+12𝐿+1=0. Therefore, 1=sin𝜃0+cos𝜃1=cos𝜃1+cos𝜃1=0,2𝐿=cos𝜃2𝐿sin𝜃2𝐿+1=cos𝜃2𝐿+cos𝜃102𝐿+1=0.(2.11) It follows that the support of {𝑛}𝑛 is in {0,1,2,,2𝐿1}. Suppose that 𝜋/2<𝜃<𝜋/2, then 00. It follows that there exists 𝑡(𝑡𝐿), such that 02𝑡10. Furthermore, if 𝑛{0,1,2,,2𝑡1}, then 𝑛=0. By the hypothesis, the theorem is proved.

Theorem 2.4 shows that any one-dimensional orthogonal low-pass wavelet filter can be constructed by a series of SUT. Suppose that an orthogonal scaling function 𝜙(𝑥) satisfies (2.1). Then the sequence {𝑛}𝑛 is a one-dimensional CQF. By Theorem 2.4, we know that {𝑛}𝑛 can be constructed by a simple one-dimensional CQF and a series of 2×2 orthogonal matrices. For a real number 𝜃, we denote 𝐺𝜃 as the 2×2 orthogonal matrix: cos𝜃sin𝜃sin𝜃cos𝜃,𝜃.(2.12) Let {𝑛}𝛾𝑛 be a simple one-dimensional CQF such that 𝑛=cos𝛾,𝑛=0,sin𝛾,𝑛=1,0,otherwise.𝛾.(2.13) For an arbitrary positive integer 𝑁, choosing 𝛾0,𝛾1,,𝛾𝑁1, we define {𝑁𝑛}𝑛 as follows: 𝑁𝑛𝑛=𝐆𝛾𝑁1𝐆𝛾𝑁2𝐆𝛾1𝑛𝛾0𝑛.(2.14) In general, the support of {𝑁𝑛}𝑛 is in [0,2𝑁1]. In other words, {𝑁𝑛}𝑛 is a length-2𝑁 filter. From the proof of Theorem 2.4, we know that the length-2𝑁 filter can be constructed by at most 𝑁1 times SUT of one-dimensional CQF.

We can prove inductively the following theorems.

Theorem 2.5. Let 𝜂=𝛾0+𝛾1++𝛾𝑁1, then 𝑖𝑁2𝑖=cos𝜂,𝑖𝑁2𝑖+1=sin𝜂.(2.15)

Theorem 2.6. Let 𝛾0+𝛾1++𝛾𝑁1𝜋=2𝑘𝜋+4,𝑘,(2.16) then the sequence {𝑁𝑛}𝑛 given in (2.14) is a one-dimensional low-pass wavelet filter.

Theorem 2.6 provides a condition for choosing 𝛾0,𝛾1,,𝛾𝑁1 such that the one-dimensional CQF {𝑁𝑛}𝑛 given in (2.14) is a one-dimensional orthogonal low-pass wavelet filter.

In addition, if the low-pass wavelet filter {𝑁𝑛}𝑛 satisfies the Cohen's condition (see [4]), then the 𝜑(𝑥) corresponding to {𝑁𝑛}𝑛 in (2.2) is an orthogonal scaling function.

By Theorem 2.6, we know that a length-2𝑁 one-dimensional low-pass wavelet filter {𝑁𝑛}𝑛 can be constructed by choosing 𝛾0,𝛾1,,𝛾𝑁1 such that condition (2.16) is satisfied. Therefore, any length-2𝑁 one-dimensional low-pass wavelet filter can be parameterized into a (𝑁1)-parameter family of wavelet system. In fact, we can give an explicit parametrization of any length-2𝑁 filter: 𝑁𝑛𝑛=𝐆𝛾𝑁1𝐆𝛾𝑁2𝐆𝛾1𝑛𝜋/4𝛾𝑁1𝛾𝑁2𝛾1𝑛,(2.17) where 𝛾1,𝛾2,,𝛾𝑁1.

Applications of one-dimensional parameterized wavelets to compression are, for example, discussed in [22, 23]. Parameterizing all possible filter coefficients that correspond to compactly supported one-dimensional orthonormal wavelets has been studied by several authors [6, 9, 2426]. We provide explicit parametrization of any length-2𝑁 filters which satisfy the necessary conditions for orthogonality in terms of SUT.

3. Construction of Two-Dimensional Wavelet Filters

3.1. SUT of Two-Dimensional CQF

To construct two-dimensional orthogonal scaling function 𝜙(x)=2𝛼2𝑏𝛼𝜙(2x𝛼)(3.1) and its associated wavelets 𝜓𝑙(x)=2𝛼2𝑑𝑙𝛼𝜙(2x𝛼),𝑙=1,2,3,(3.2) we need construct sequences {𝑏𝛼}𝛼2 and {𝑑𝑙𝛼}𝛼2 such that (see [10, 12, 13], etc.) 𝛼2𝑏𝛼=2,(3.3)𝛼2𝑏𝛼𝑏𝛼+2𝛽=𝛿0𝛽,(3.4)𝑖,𝑗𝑏(2𝑖,2𝑗)=𝑖,𝑗𝑏(2𝑖+1,2𝑗)=𝑖,𝑗𝑏(2𝑖,2𝑗+1)=𝑖,𝑗𝑏(2𝑖+1,2𝑗+1),(3.5)𝛼2𝑑𝑗1𝛼𝑑𝑗2𝛼+2𝛽=𝛿𝑗1,𝑗2𝛿0,𝛽,(3.6)𝛼2𝑏𝛼𝑑𝑗𝛼+2𝛽=0,(3.7) where 𝑗,𝑗1,𝑗2=1,2,3 and 𝛽2.

The sequence {𝑏𝛼}𝛼2, which satisfies (3.4), is called a two-dimensional CQF. If {𝑏𝛼}𝛼2 satisfies (3.3), (3.4), and (3.5) simultaneously, we call it a two-dimensional low-pass wavelet filter. The sequences {𝑑𝑗𝛼}𝛼2(𝑗=1,2,3), which satisfy (3.6) and (3.7), are called two-dimensional high-pass wavelet filters.

For an arbitrary real-valued sequence {𝑙𝛼}𝛼22(2), we define Δ=𝛼𝛼2,𝑙𝛼,0(3.8) and Δ is called the support set of {𝑙𝛼}𝛼22(2). If Δ is finite, we call {𝑙𝛼}𝛼22(2) finite supported sequence. As aforementioned, the sequences we consider are real-valued and finite supported (FIR).

We note that any sequence {𝑏𝛼}𝛼2 can be split into 4 disjoint subsets 𝑏(2𝑖,2𝑗),𝑏𝑖,𝑗(2𝑖,2𝑗+1),𝑏𝑖,𝑗(2𝑖+1,2𝑗),𝑏𝑖,𝑗(2𝑖+1,2𝑗+1).𝑖,𝑗(3.9)

Definition 3.1. Let 𝑈 be an arbitrary 4×4 orthogonal matrix, and let {𝑏𝛼}𝛼2 be an FIR. For integers𝐴𝑠,𝐵𝑠(𝑠=1,2,3,4) and for all 𝑖,𝑗, we set Γ𝑖,𝑗=̃𝑏(2𝑖,2𝑗)̃𝑏(2𝑖,2𝑗+1)̃𝑏(2𝑖+1,2𝑗)̃𝑏(2𝑖+1,2𝑗+1))𝑇,Γ𝑖,𝑗(𝐴1,𝐵1,𝐴2,𝐵2,𝐴3,𝐵3,𝐴4,𝐵4)=𝑏(2𝑖2𝐴1,2𝑗2𝐵1)𝑏(2𝑖2𝐴2,2𝑗2𝐵2+1)𝑏(2𝑖2𝐴3+1,2𝑗2𝐵3)𝑏(2𝑖2𝐴4+1,2𝑗2𝐵4+1)𝑇.(3.10) Then {̃𝑏𝛼}𝛼2, which is defined as follows: Γ𝑖,𝑗=Γ𝑖,𝑗(𝐴1,𝐵1,𝐴2,𝐵2,𝐴3,𝐵3,𝐴4,𝐵4)𝑈(3.11) is called the two-dimensional SUT of {𝑏𝛼}𝛼2.

Lemma 3.2. If {𝑏𝛼}𝛼2 is a two-dimensional CQF and {̃𝑏𝛼}𝛼2 is given by (3.11), then {̃𝑏𝛼}𝛼2 is also a two-dimensional CQF.

Proof. By directly calculating, we can prove that {̃𝑏𝛼}𝛼2 satisfies the following equation: 𝛼2̃𝑏𝛼̃𝑏𝛼+2𝛽=𝛿0,𝛽,𝛽2.(3.12) This completes the proof.

If the new two-dimensional CQF is a low-pass wavelet filter, then it is worthwhile to construct the associated high-pass wavelet filters. Now we provide a result of it.

Lemma 3.3. Suppose that {𝑏𝛼}𝛼2 is a two-dimensional CQF, {𝑑𝑘𝛼}𝛼2(𝑘=1,2,3) satisfy (3.6) and (3.7), {̃𝑏𝛼}𝛼2 is a two-dimensional CQF obtained by SUT of {𝑏𝛼}𝛼2, then {𝑑𝑘𝛼}𝛼2, which are derived by SUT of {𝑑𝑘𝛼}𝛼2(𝑘=1,2,3), satisfy (3.6) and (3.7) simultaneously.

It can be proved by direct calculation, so we omit the proof.

Definition 3.4. (i) {̃𝑏}𝛼2 is called the SUT0 of {𝑏𝛼}𝛼2 if one chooses 𝐴𝑖=𝐵𝑖=0(𝑖=1,2,3,4)(3.13) in (3.11). This transform is denoted by {̃𝑏𝛼}𝛼2=0𝐔{𝑏𝛼}𝛼2.
(ii) {̃𝑏}𝛼2 is called the SUTT1 of {𝑏𝛼}𝛼2 if one chooses 𝐵2=𝐵4=1,𝐵1=𝐵3=𝐴𝑖=0(𝑖=1,2,3,4)(3.14) in (3.11). This transform is denoted by {̃𝑏𝛼}𝛼2=𝑇1𝐔{𝑏𝛼}𝛼2.{̃𝑏}𝛼2 is called the SUTT2 of {𝑏𝛼}𝛼2 if one chooses 𝐴3=𝐴4=1,𝐴1=𝐴2=𝐵𝑖=0(𝑖=1,2,3,4).(3.15) This transform is denoted by {̃𝑏𝛼}𝛼2=𝑇2𝐔{𝑏𝛼}𝛼2.
(iii) {̃𝑏}𝛼2 is called the SUT1 of {𝑏𝛼}𝛼2 if one chooses 𝐴2=𝐴4=1,𝐴1=𝐴3=𝐵𝑖=0(𝑖=1,2,3,4)(3.16) in (3.11). This transform is denoted by {̃𝑏𝛼}𝛼2=1𝐔{𝑏𝛼}𝛼2.{̃𝑏}𝛼2 is called the SUT2 of {𝑏𝛼}𝛼2 if we choose 𝐵3=𝐵4=1,𝐵1=𝐵2=𝐴𝑖=0(𝑖=1,2,3,4).(3.17) This transform is denoted by {̃𝑏𝛼}𝛼2=2𝐔{𝑏𝛼}𝛼2.

For a two-dimensional CQF, when we choose different orthogonal matrices, many new two-dimensional CQFs can be obtained. It is obvious that, after SUT0, the support of the new two-dimensional CQF does not change. But it is different for SUTT1, SUTT2, SUT1, and SUT2. For example, the support of the two-dimensional CQF 𝑏𝛼=14if𝛼=(0,0)or(1,1),34if𝛼=(1,0)or(0,1),0otherwise(3.18) is {(0,0),(0,1),(1,0),(1,1)}, and the support of the filter {̃𝑏𝛼}𝛼2=1𝐔{𝑏𝛼}𝛼2 is {(0,0),(0,1),(1,0),(1,1),(2,0),(2,1),(3,0),(3,1)}([0,3]×[0,1]2), where 𝑈=diag(𝐻,𝐻) and 1𝐻=2323212.(3.19) In general, for integers 𝑁,𝑀, if the support of {𝑏𝛼}𝛼2 is in [0,2𝑁1]×[0,2𝑀1]2, after SUT1 (or SUT2) the support of the new filter is in [0,2𝑁+1]×[0,2𝑀1]2 (or in [0,2𝑁1]×[0,2𝑀+1]2). If the support of {𝑏𝛼}𝛼2 is {(0,0),(1,0),(0,1),(1,1)}, we call {𝑏𝛼}𝛼2 a simple two-dimensional filter.

We will adopt the following notations in the rest of this paper. For arbitrary 𝜉0,𝜆0,𝜉,𝜆𝑅, let {𝛼}𝜆0,𝜉0𝛼2 be the FIR defined as follows: 𝛼=cos𝜉0cos𝜆0,𝛼=(0,0);cos𝜉0sin𝜆0,𝛼=(0,1);sin𝜉0cos𝜆0,𝛼=(1,0);sin𝜉0sin𝜆0,𝛼=(1,1);0,otherwise.(3.20) Furthermore, let {𝒟𝑘𝛼}𝜆0,𝜉0𝛼2(𝑘=1,2,3) be the filters as follows 𝒟1𝛼=cos𝜉0sin𝜆0,𝛼=(0,0);cos𝜉0cos𝜆0,𝛼=(0,1);sin𝜉0sin𝜆0,𝛼=(1,0);sin𝜉0cos𝜆0𝒟,𝛼=(1,1);0,otherwise,2𝛼=sin𝜉0cos𝜆0,𝛼=(0,0);sin𝜉0sin𝜆0,𝛼=(0,1);cos𝜉0cos𝜆0,𝛼=(1,0);cos𝜉0sin𝜆0𝒟,𝛼=(1,1);0,otherwise,(3.21)3𝛼=sin𝜉0sin𝜆0,𝛼=(0,0);sin𝜉0cos𝜆0,𝛼=(0,1);cos𝜉0sin𝜆0,𝛼=(1,0);cos𝜉0cos𝜆0,𝛼=(1,1);0,otherwise,(3.22) and let 𝑈𝜆,𝑈𝜉 be orthogonal matrices such that 𝑈𝜆=,𝑈cos𝜆sin𝜆00sin𝜆cos𝜆0000cos𝜆sin𝜆00sin𝜆cos𝜆𝜉=,cos𝜉0sin𝜉00cos𝜉0sin𝜉sin𝜉0cos𝜉00sin𝜉0cos𝜉(3.23) respectively. Then {𝛼}𝜆0,𝜉0𝛼2 is a simple two-dimensional CQF, and {𝒟𝑘𝛼}𝜆0,𝜉0𝛼2 satisfy (3.6) and (3.7).

From the special two-dimensional CQF {𝛼}𝜆0,𝜉0𝛼2, by SUTT1, SUTT2, SUT1, SUT2, and the matrices 𝑈𝜆,𝑈𝜉, we can construct some new two-dimensional CQFs.

3.2. Tensor-Product Wavelet Filters

In this subsection, we will show that all tensor-product wavelet filters can be constructed by SUTT1 and SUTT2.

A two-dimensional low-pass wavelet filters {𝑏𝛼}𝛼2 is called tensor-product wavelet filter if it satisfies the following equations: 𝑏(2𝑖,2𝑗)=2𝑖2𝑗,𝑏(2𝑖,2𝑗+1)=2𝑖2𝑗+1,𝑏(2𝑖+1,2𝑗)=2𝑖+12𝑗,𝑏(2𝑖+1,2𝑗+1)=2𝑖+12𝑗+1,𝑖,𝑗,(3.24) where {𝑖}𝑖, {𝑖}𝑖 are one-dimensional orthogonal low-pass wavelet filters.

Theorem 3.5. If {𝑏𝛼}𝛼2 is a tensor-product low-pass wavelet filter, then, it can be constructed by SUTT1 and SUTT2 from {𝛼}𝜆0,𝜉0𝛼2.

Proof. Forall𝑖,𝑗, it follows from (3.24) that 𝑏(2𝑖,2𝑗),𝑏(2𝑖,2𝑗+1),𝑏(2𝑖+1,2𝑗),𝑏(2𝑖+1,2𝑗+1)=2𝑖,2𝑖,2𝑖+1,2𝑖+12𝑗00002𝑗+100002𝑗00002𝑗+1.(3.25) Suppose that {𝑖}𝑖 is a length-2𝑁 one-dimensional filter. Then there exist 𝜆0,,𝜆𝑁2,𝜆𝑁1𝑅 such that 𝜆0++𝜆𝑁2+𝜆𝑁1=𝜋/4, and 𝑛𝑛=𝐆𝜆𝑁1𝐆𝜆𝑁2𝐆𝜆1𝑛𝜆0𝑛.(3.26) We denote 𝑛𝑁1𝑛=𝐆𝜆𝑁2𝐆𝜆1𝑛𝜆0𝑛.(3.27) Let {𝑏𝛼}𝑁1𝛼2 be two-dimensional filter such that 𝑏𝑁1(2𝑖,2𝑗)=2𝑖𝑁12𝑗,𝑏𝑁1(2𝑖,2𝑗+1)=2𝑖𝑁12𝑗+1,𝑏𝑁1(2𝑖+1,2𝑗)=2𝑖+1𝑁12𝑗,𝑏𝑁1(2𝑖+1,2𝑗+1)=2𝑖+1𝑁12𝑗+1,𝑖,𝑗.(3.28) It follows that 𝑏(2𝑖,2𝑗),𝑏(2𝑖,2𝑗+1),𝑏(2𝑖+1,2𝑗),𝑏(2𝑖+1,2𝑗+1)=2𝑖,2𝑖,2𝑖+1,2𝑖+12𝑗00002𝑗+100002𝑗00002𝑗+1=2𝑖,2𝑖,2𝑖+1,2𝑖+1𝑁12𝑗0000𝑁12𝑗10000𝑁12𝑗0000𝑁12𝑗1×cos𝜆𝑁1sin𝜆𝑁100sin𝜆𝑁1cos𝜆𝑁10000cos𝜆𝑁1sin𝜆𝑁100sin𝜆𝑁1cos𝜆𝑁1=𝑏𝑁1(2𝑖,2𝑗),𝑏𝑁1(2𝑖,2𝑗1),𝑏𝑁1(2𝑖+1,2𝑗),𝑏𝑁1(2𝑖+1,2𝑗1)𝑈𝜆𝑁1.(3.29) Hence, {𝑏𝛼}𝛼2 can be constructed by SUTT1 of {𝑏𝛼}𝑁1𝛼2.
It can be proved inductively that {𝑏𝛼}𝛼2 can be constructed by a series of SUTT1 from {𝑏𝛼}0𝛼2, and {𝑏𝛼}0𝛼2 can be constructed by a series of SUTT2 from {𝛼}𝜆0,𝜉0𝛼2, where {𝑏𝛼}0𝛼2 denotes the filter 𝑏0(2𝑖,2𝑗)=2𝑖2𝑗,𝑏0(2𝑖,2𝑗+1)=2𝑖2𝑗+1,𝑏0(2𝑖+1,2𝑗)=2𝑖+12𝑗,𝑏0(2𝑖+1,2𝑗+1)=2𝑖+12𝑗+1,𝑖,𝑗.(3.30) Therefore, any tensor-product two-dimensional low-pass wavelet filters can be constructed by SUTT1 and SUTT2 from {𝛼}𝜆0,𝜉0𝛼2.

3.3. Two-Dimensional Wavelet Filters in Terms of SUT1 and SUT2

From now on, we give a method of construction of two-dimensional orthogonal wavelet filters from a simple two-dimensional CQF by SUT1 and SUT2.

For arbitrary positive integers 𝑁 and 𝑀, choosing 𝜆0,𝜆1,,𝜆𝑁1,𝜉0,𝜉1,,𝜉𝑀1, {𝑏𝛼𝑁,𝑀}𝛼2 is defined as 𝑏𝛼𝑁,𝑀𝛼2=𝜀𝑛𝑁+𝑀2𝑈𝑛𝑁+𝑀2𝜀𝑛𝑁+𝑀3𝑈𝑛𝑁+𝑀3𝜀𝑛1𝑈𝑛1𝛼𝜆0,𝜉0𝛼2,(3.31) where 𝑛1,𝑛2,,𝑛𝑁+𝑀2{𝜆1,𝜆2,,𝜆𝑁1,𝜉1,𝜉2,,𝜉𝑀1} and 𝜀𝑛𝑗=1,if𝑛𝑗𝜆1,𝜆2,,𝜆𝑁1,2,if𝑛𝑗𝜉1,𝜉2,,𝜉𝑀1.(3.32) Then {𝑏𝛼𝑁,𝑀}𝛼2 is a two-dimensional CQF, and its support is in [0,2𝑁1]×[0,2𝑀1]2. We are now in a position to draw some conditions on choosing 𝜆0,𝜆1,,𝜆𝑁1,𝜉0,𝜉1,,𝜉𝑀1, such that {𝑏𝛼𝑁,𝑀}𝛼2 is a two-dimensional low-pass wavelet filter.

Theorem 3.6. Let 𝜂=𝜆0+𝜆1++𝜆𝑁1,̃𝜂=𝜉0+𝜉1++𝜉𝑀1.Then 𝑖,𝑗𝑏𝑁,𝑀(2𝑖,2𝑗)=cos𝜂cos̃𝜂,𝑖,𝑗𝑏𝑁,𝑀(2𝑖,2𝑗+1)=sin𝜂cos̃𝜂,(3.33)𝑖,𝑗𝑏𝑁,𝑀(2𝑖+1,2𝑗)=cos𝜂siñ𝜂,𝑖,𝑗𝑏𝑁,𝑀(2𝑖+1,2𝑗+1)=sin𝜂siñ𝜂.(3.34)

Proof. It can be proved inductively. For the case 𝑁=𝑀=1, it is obviously true. Assume that it is true for the case 𝑁𝑘1,𝑀𝑘2(𝑘11,𝑘21,𝑘1,𝑘2); now we prove that it is true for the case 𝑁=𝑘1+1,𝑀=𝑘2.
Let 𝑠 be the integer such that 𝑛𝑠+1{𝜆1,𝜆2,,𝜆𝑘1} and 𝑛𝑠+𝑡{𝜉1,𝜉2,,𝜉𝑘2}(𝑡>1,𝑡). Suppose that 𝑖1,𝑖2,,𝑖𝑢𝑠𝑛𝑖1,𝑛𝑖2,,𝑛𝑖𝑢𝜆1,𝜆2,,𝜆𝑘1,𝑗1,𝑗2,,𝑗𝑣𝑠𝑛𝑗1,𝑛𝑗2,,𝑛𝑗𝑣𝜉1,𝜉2,,𝜉𝑘1.(3.35) It follows that 𝑢=𝑘1 and 𝑣𝑘21. Let 𝜂𝑠=𝜆0+𝑛𝑖1+𝑛𝑖2++𝑛𝑖𝑢,̃𝜂𝑠=𝜉0+𝑛𝑗1+𝑛𝑗2++𝑛𝑗𝑣,𝑏𝑠𝛼𝛼2=𝜀𝑛𝑠𝑈𝑛𝑠𝜀𝑛𝑠1𝑈𝑛𝑠1𝜀𝑛1𝑈𝑛1𝛼𝜆0,𝜉0𝛼2;(3.36) it follows that 𝑖,𝑗𝑏𝑠(2𝑖,2𝑗)=cos𝜂𝑠cos̃𝜂𝑠,𝑖,𝑗𝑏𝑠(2𝑖,2𝑗+1)=sin𝜂𝑠cos̃𝜂𝑠,𝑖,𝑗𝑏𝑠(2𝑖+1,2𝑗)=cos𝜂𝑠siñ𝜂𝑠,𝑖,𝑗𝑏𝑠(2𝑖+1,2𝑗+1)=sin𝜂𝑠siñ𝜂𝑠.(3.37) Let {𝑏𝛼𝑠+1}𝛼2=𝜀𝑛𝑠+1𝑈𝑛𝑠+1{𝑏𝑠𝛼}𝛼2. Since 𝑛𝑠+1{𝜆1,𝜆2,,𝜆𝑘1}, then 𝑏𝑠+1(2𝑖,2𝑗)=cos𝑛𝑠+1𝑏𝑠(2𝑖,2𝑗)sin𝑛𝑠+1𝑏𝑠(2𝑖2,2𝑗+1),𝑏𝑠+1(2𝑖,2𝑗+1)=sin𝑛𝑠+1𝑏𝑠(2𝑖,2𝑗)+cos𝑛𝑠+1𝑏𝑠(2𝑖2,2𝑗+1),𝑏𝑠+1(2𝑖+1,2𝑗)=cos𝑛𝑠+1𝑏𝑠(2𝑖+1,2𝑗)sin𝑛𝑠+1𝑏𝑠(2𝑖1,2𝑗+1),𝑏𝑠+1(2𝑖+1,2𝑗+1)=sin𝑛𝑠+1𝑏𝑠(2𝑖+1,2𝑗)+cos𝑛𝑠+1𝑏𝑠(2𝑖1,2𝑗+1).(3.38) Therefore, 𝑖,𝑗𝑏𝑠+1(2𝑖,2𝑗)𝜂=cos𝑠+𝑛𝑠+1cos̃𝜂𝑠;𝑖,𝑗𝑏𝑠+1(2𝑖,2𝑗+1)𝜂=sin𝑠+𝑛𝑠+1cos̃𝜂𝑠;𝑖,𝑗𝑏𝑠+1(2𝑖+1,2𝑗)𝜂=cos𝑠+𝑛𝑠+1siñ𝜂𝑠;𝑖,𝑗𝑏𝑠+1(2𝑖+1,2𝑗+1)𝜂=sin𝑠+𝑛𝑠+1siñ𝜂𝑠.(3.39) For 𝑡>1, let 𝑏𝛼𝑠+𝑡𝛼2=𝜀𝑛𝑠+𝑡𝑈𝑛𝑠+𝑡𝜀𝑛𝑠+1𝑈𝑛𝑠+1𝑏𝑠𝛼𝛼2.(3.40) It follows from 𝑛𝑠+𝑡{𝜉1,𝜉2,,𝜉𝑘21} that 𝑏𝑠+𝑡(2𝑖,2𝑗)=cos𝑛𝑠+𝑡𝑏𝑠+𝑡1(2𝑖,2𝑗)sin𝑛𝑠+𝑡𝑏𝑠+𝑡1(2𝑖+1,2𝑗2),𝑏𝑠+𝑡(2𝑖,2𝑗+1)=cos𝑛𝑠+𝑡𝑏𝑠+𝑡1(2𝑖,2𝑗+1)sin𝑛𝑠+𝑡𝑏𝑠+𝑡1(2𝑖+1,2𝑗1),𝑏𝑠+𝑡(2𝑖+1,2𝑗)=sin𝑛𝑠+𝑡𝑏𝑠+𝑡1(2𝑖,2𝑗)+cos𝑛𝑠+𝑡𝑏𝑠+𝑡1(2𝑖+1,2𝑗2),𝑏𝑠+𝑡(2𝑖+1,2𝑗+1)=sin𝑛𝑠+𝑡𝑏𝑠+𝑡1(2𝑖,2𝑗+1)+cos𝑛𝑠+𝑡𝑏𝑠+𝑡1(2𝑖+1,2𝑗1).(3.41) Therefore, 𝑖,𝑗𝑏𝑠+𝑡(2𝑖,2𝑗)=cos𝜂𝑠+𝑡1cos̃𝜂𝑠+𝑡1+𝑛𝑠+𝑡;𝑖,𝑗𝑏𝑠+𝑡(2𝑖,2𝑗+1)=sin𝜂𝑠+𝑡1cos̃𝜂𝑠+𝑡1+𝑛𝑠+𝑡;𝑖,𝑗𝑏𝑠+𝑡(2𝑖+1,2𝑗)=cos𝜂𝑠+𝑡1siñ𝜂𝑠+𝑡1+𝑛𝑠+𝑡;𝑖,𝑗𝑏𝑠+𝑡(2𝑖+1,2𝑗+1)=sin𝜂𝑠+𝑡1siñ𝜂𝑠+𝑡1+𝑛𝑠+𝑡,(3.42) where 𝜂𝑠+𝑡1=𝜆0+𝜆1++𝜆𝑘1,̃𝜂𝑠+𝑡1=𝜉0+𝑛𝑗1++𝑛𝑗𝑣+𝑛𝑠+2+𝑛𝑠+𝑡1.(3.43) Namely, it is true for the case 𝑁=𝑘1+1,𝑀=𝑘2.
Similarly for the case 𝑁=𝑘1,𝑀=𝑘2+1. This completes the proof.

Now we provide the condition on choosing 𝜆0,𝜆1,,𝜆𝑁1,𝜉0,𝜉1,,𝜉𝑀1, such that {𝑏𝛼𝑁,𝑀}𝛼2 is a two-dimensional low-pass wavelet filter.

Theorem 3.7. If there exists integers 𝑛1 and 𝑛2, such that 𝜂=𝜆0+𝜆1++𝜆𝑁1=2𝑛1𝜋𝜋+4,(3.44)̃𝜂=𝜉0+𝜉1++𝜉𝑀1=2𝑛2𝜋𝜋+4,(3.45) then the sequence {𝑏𝛼𝑁,𝑀}𝛼2 constructed in (3.31) is a two-dimensional low-pass wavelet filter.

Proof. It follows from Theorem 3.6 that 𝑖,𝑗𝑏(2𝑖,2𝑗)1=cos𝜂cos̃𝜂=2,𝑖,𝑗𝑏(2𝑖,2𝑗+1)1=sin𝜂cos̃𝜂=2,𝑖,𝑗𝑏(2𝑖+1,2𝑗)1=cos𝜂siñ𝜂=2,𝑖,𝑗𝑏(2𝑖+1,2𝑗+1)1=sin𝜂siñ𝜂=2.(3.46) Therefore, {𝑏𝛼𝑁,𝑀}𝛼2 satisfies conditions (3.3) and (3.5), then it is a two-dimensional low-pass wavelet filter.

Corollary 3.8. If {𝑏𝛼𝑁,𝑀}𝛼2 as given in (3.31) is a low-pass wavelet filter, then 𝑑𝛼𝑘,𝑁,𝑀𝛼2=𝜀𝑛𝑁+𝑀2𝑈𝑛𝑁+𝑀2𝜀𝑛𝑁+𝑀3𝑈𝑛𝑁+𝑀3𝜀𝑛1𝑈𝑛1𝒟𝑘𝛼𝜆0,𝜉0𝛼2(3.47) are the high-pass filters associated with {𝑏𝛼𝑁,𝑀}𝛼2, where 𝑘=1,2,3.

Remark 3.9. We can choose other orthogonal matrices than U𝜆 and U𝜉 such as 𝑈𝜆,̃𝜆=̃̃𝜆̃̃𝜆,𝑈cos𝜆sin𝜆00sin𝜆cos𝜆0000cos𝜆sin00sin𝜆cos𝜉,̃𝜉=̃̃𝜉̃̃𝜉,cos𝜉0sin𝜉00cos𝜉0sinsin𝜉0cos𝜉00sin𝜉0cos(3.48) where ̃̃𝜆,𝜆,𝜉,𝜉; but conditions in Theorem 3.7 should be changed.

Example 3.10. We choose 𝑁=2,𝑀=2 in (3.31). For 𝜉0,𝜉1,𝜆0,𝜆1, define {𝑏𝛼}𝛼2 as follows: 𝑏𝛼𝛼2=2𝑈𝜉11𝑈𝜆1𝛼𝜆0,𝜉0𝛼2.(3.49)

By choosing 𝜉0,𝜉1,𝜆0,𝜆1 such that 𝜆0+𝜆1=𝜉0+𝜉1=𝜋/4, we can get many two-dimensional low-pass wavelet filters and their corresponding high-pass wavelet filters. For instance, set 𝜉1=2.254190, 𝜉0=𝜋/4𝜉1, 𝜆1=4.357946, 𝜆0=𝜋/4𝜆1. By (3.31) and (3.47), we can get a nonseparable orthogonal low-pass wavelet filter (see Table 1) and its associated high-pass wavelet filters (Tables 2, 3, and 4). Figure 1 shows that the high frequency sub-bands by the derived filter can reveal more features than that by the commonly used tensor-product wavelet filter.


𝑏 ( 𝑖 , 𝑗 ) 𝑗 = 0 𝑗 = 1 𝑗 = 2 𝑗 = 3

𝑖 = 0 −0.020275 −0.054787 0.243250 0.657306
𝑖 = 1 0.024899 0.067282 0.198076 0.535237
𝑖 = 2 −0.025190 0.009322 0.302214 −0.111841
𝑖 = 3 0.030935 −0.011448 0.246090 −0.091071


𝑑 1 ( 𝑖 , 𝑗 ) 𝑗 = 0 𝑗 = 1 𝑗 = 2 𝑗 = 3

𝑖 = 0 0.009322 0.025190 −0.111841 −0.302214
𝑖 = 1 −0.011448 −0.030935 −0.091071 −0.246090
𝑖 = 2 −0.054787 0.020275 0.657306 −0.243250
𝑖 = 3 0.067282 −0.024899 0.535237 −0.198076


𝑑 2 ( 𝑖 , 𝑗 ) 𝑗 = 0 𝑗 = 1 𝑗 = 2 𝑗 = 3

𝑖 = 0 0.198076 0.535237 0.024899 0.067282
𝑖 = 1 −0.243250 −0.657306 0.020275 0.054787
𝑖 = 2 0.246090 −0.091071 0.030935−0.011448
𝑖 = 3 −0.302214 0.111841 0.025190 −0.009322


𝑑 3 ( 𝑖 , 𝑗 ) 𝑗 = 0 𝑗 = 1 𝑗 = 2 𝑗 = 3

𝑖 = 0 −0.091071 −0.246090 −0.011448 −0.030935
𝑖 = 1 0.111841 0.302214 −0.009322 −0.025190
𝑖 = 2 0.535237 −0.198076 0.067282 −0.024899
𝑖 = 3 −0.657306 0.243250 0.054787 −0.020275

Remark 3.11. By Section 3.2, we know that any two-dimensional tensor-product orthogonal wavelet filters can be constructed by SUTT1 and SUTT2. By Example 3.10, we know that, by SUT1 and SUT2, nonseparable wavelet filters can be achieved. Therefore, the construction of two-dimensional wavelet filters in terms of SUT of two-dimensional CQF is a generalization of the construction of separable orthogonal wavelet filters. Furthermore, from (3.31) and (3.47), we can see that our construction is a parametrization method.

4. Conclusion

SUT of CQF is introduced in this paper. In terms of SUT of one-dimensional CQF, any one-dimensional orthogonal wavelet filters with dilation factor 2 can be given in explicit expression. The SUT of two-dimensional CQF is applied to the construction of two-dimensional orthogonal wavelet filters, and a parametrization method is presented. The selection of the parameters is not restricted by any implicit condition. Tensor-product wavelet filters are only special case of this method. It provides more ways to randomly generate perfect reconstruction filters.

Our method provides many possible choices for the parameters. But what is a good choice of the parameters? Should any restriction on the choice of the parameters imply certain properties? Characteristics of SUT should be deeply studied.

Acknowledgments

The authors would like to thank the anonymous reviewers and the associate editor for their valuable comments and suggestions to improve the presentation of this paper. This work was supported in part by the National Science Foundation of China under Grant 60973157. This work was also supported in part by the National Science Foundation of Jiangsu Province Education Department under Grant no. 08KJB520004, and in part by Philosophy and Social Sciences Planning Project of Beijing Municipal Commission of Education under Grant no. SM201010011002.

References

  1. I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Communications on Pure and Applied Mathematics, vol. 41, no. 7, pp. 909–996, 1988. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  2. M. Vetterli, “Wavelets, approximation, and compression,” IEEE Signal Processing Magazine, vol. 18, no. 5, pp. 59–73, 2001. View at: Publisher Site | Google Scholar
  3. A. Cohen, I. Daubechies, and J. C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Communications on Pure and Applied Mathematics, vol. 45, no. 5, pp. 485–560, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  4. I. Daubechies, Ten Lectures on Wavelets, vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1992. View at: Zentralblatt MATH
  5. P. N. Heller, “Rank m wavelets with n vanishing moments,” SIAM Journal on Matrix Analysis and Applications, vol. 16, no. 2, pp. 502–519, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  6. G. Regensburger, “Parametrizing compactly supported orthonormal wavelets by discrete moments,” Applicable Algebra in Engineering, Communication and Computing, vol. 18, no. 6, pp. 583–601, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. H. L. Resnikoff, J. Tian, and R. O. Wells, Jr., “Biorthogonal wavelet space: parametrization and factorization,” SIAM Journal on Mathematical Analysis, vol. 33, no. 1, pp. 194–215, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  8. W. Sweldens, “The lifting scheme: a custom-design construction of biorthogonal wavelets,” Applied and Computational Harmonic Analysis, vol. 3, no. 2, pp. 186–200, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  9. P. P. Vaidyananathan, Multirate Systems and Filter Banks, Prentice-Hall, Cliffs, NJ, USA, 1993.
  10. A. Ayache, “Some methods for constructing nonseparable, orthonormal, compactly supported wavelet bases,” Applied and Computational Harmonic Analysis, vol. 10, no. 1, pp. 99–111, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  11. E. Belogay and Y. Wang, “Arbitrarily smooth orthogonal nonseparable wavelets in R2,” SIAM Journal on Mathematical Analysis, vol. 30, no. 3, pp. 678–697, 1999. View at: Publisher Site | R2&author=E. Belogay &author=Y. Wang&publication_year=1999" target="_blank">Google Scholar | Zentralblatt MATH
  12. W. He and M. J. Lai, “Examples of bivariate nonseparable compactly supported orthonormal continuous wavelets,” IEEE Transactions on Image Processing, vol. 9, no. 5, pp. 949–953, 2000. View at: Publisher Site | Google Scholar
  13. M. J. Lai, “Construction of multivariate compactly supported orthonormal wavelets,” Advances in Computational Mathematics, vol. 25, no. 1–3, pp. 41–56, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  14. A. Cohen and I. Daubechies, “Nonseparable bidimensional wavelet bases,” Revista Matemática Iberoamericana, vol. 9, no. 1, pp. 51–137, 1993. View at: Google Scholar
  15. J. Kovačević and M. Vetterli, “Nonseparable multidimensional perfect reconstruction filter banks,” IEEE Transactions on Information Theory, vol. 38, no. 2, part 2, pp. 533–555, 1992. View at: Publisher Site | Google Scholar
  16. G. Strang and T. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press, Wellesley, Mass, USA, 1996.
  17. Y. W. Wang, J. F. Doherty, and R. E. Van Dyck, “A waveletbased watermarking algorithm for ownership verification of digital image,” IEEE Transactions on Image Processing, vol. 11, no. 2, pp. 77–88, 2002. View at: Google Scholar
  18. W. Dietl, P. Meerwald, and A. Uhl, “Protection of waveletbased watermarking systems using filter parametrization,” Signal Processing, vol. 83, no. 10, pp. 2095–2116, 2003. View at: Google Scholar
  19. Z. Huang and Z. Jiang, “Image ownership verification via private pattern and watermarking wavelet filters,” in Proceedings of the 7th Digital Image Computing: Techniques and Applications, C. Sun, H. Talbot, S. Ourselin, and T. Adriaansen, Eds., pp. 801–810, Sydney, Australia, December 2003. View at: Google Scholar
  20. P. Meerwald and A. Uhl, “A survey of wavelet-domain watermarking algorithms,” in Electronic Imaging, Security and Watermarking of Multimedia Contents III, P. W. Wong and E. J. Delp, Eds., vol. 4314 of Proceedings of SPIE, San Jose, Calif, USA, January 2001. View at: Google Scholar
  21. W. M. Lawton, “Necessary and sufficient conditions for constructing orthonormal wavelet bases,” Journal of Mathematical Physics, vol. 32, no. 1, pp. 57–61, 1991. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  22. J. M. Hereford and D. W. Roach, “Image compression using parameterized wavelets with feedback,” in Independent Component Analyses, Wavelets, and Neural Networks, vol. 5102 of Proceedings of SPIE, pp. 267–277, Orlando, Fla, USA, April 2003. View at: Google Scholar
  23. G. Regensburger and O. Scherzer, “Symbolic computation for moments and filter coefficients of scaling functions,” Annals of Combinatorics, vol. 9, no. 2, pp. 223–243, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  24. M. J. Lai and D. Roach, “Parameterizations of univariate orthogonal wavelets with short support,” in Approximation Theory X: Wavelets, Splines, and Applications, J. Stoeckler, C. K. Chui, and L. L. Schumaker, Eds., pp. 369–384, Vanderbilt University Press, Nashville, Tenn, USA, 2002. View at: Google Scholar | Zentralblatt MATH
  25. J. P. Li and Y. Y. Tang, “General analytic construction for wavelet lowpassed filters,” in Proceedings of the 2nd International Conference on Wavelet Analysis and its Applications (WAA '01), pp. 314–320, Hong Kong, China, December 2001. View at: Google Scholar
  26. L. Shen, H. H. Tan, and J. Y. Tham, “Symmetric-antisymmetric orthonormal multiwavelets and related scalar wavelets,” Applied and Computational Harmonic Analysis, vol. 8, no. 3, pp. 258–279, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH

Copyright © 2011 Fei Li and Jianwei Yang. 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.

581 Views | 376 Downloads | 0 Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder