Shift Unitary Transform for Constructing Two-Dimensional Wavelet Filters

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, 3–9]. But, there still exists many open problems for the construction of multidimensional wavelets ([10–13], 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 ([14–16], 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 Kova𝑐̆ević [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 [17–20]). 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 [17–20], 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 [17–20].

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 𝐴^𝑇 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)=√2_𝑘∈ℤℎ_𝑘𝜑(2𝑥−𝑘),(2.1) we need construct sequences {ℎ_𝑘}_𝑘∈ℤ such that (see [4, 21] and many others) _𝑘_∈ℤℎ_𝑘=√2,(2.2)_𝑘_∈ℤℎ_2𝑘=_𝑘∈ℤℎ_2𝑘+1,(2.3)_𝑘_∈ℤℎ_𝑘ℎ_𝑘+2𝛼=𝛿_0,𝛼,𝛼∈ℤ,(2.4) where 𝛿 denotes the Kronecker delta 𝛿_𝛼,𝛽=01,,𝛼𝛼≠=𝛽𝛽,.𝛼,𝛽∈ℤ.(2.5) The sequence {ℎ_𝑘}_𝑘∈ℤ which satisfies (2.4) is called a one-dimensional CQF, and the sequence {ℎ_𝑘}_𝑘∈ℤ, which satisfies (2.2), (2.3), (2.4) simultaneously, is called one-dimensional orthogonal low-pass wavelet filter.

Definition 2.1. Let {ℎ_𝑛}_𝑛∈ℤ be a one-dimensional CQF and 𝐺 an arbitrary 2×2 orthogonal matrix. {ℎ_𝑛}_𝑛∈ℤ is called the shift unitary transform (SUT) of {ℎ_𝑛}_𝑛∈ℤby 𝐺 if {ℎ_𝑛}_𝑛∈ℤ satisfies Υ_𝑖=Λ_𝑖𝐺,𝑖∈ℤ,(2.6) and {ℎ^𝑛}_𝑛∈ℤ is called the inverse shift unitary transform (ISUT) of {ℎ_𝑛}_𝑛∈ℤ by 𝐺 if {ℎ^𝑛}_𝑛∈ℤ satisfies Λ^𝑖=Υ_𝑖𝐺,𝑖∈ℤ,(2.7) where for any 𝑖∈𝑍, Υ_𝑖=ℎ^2𝑖ℎ_2𝑖+1^𝑇,Λ_𝑖=ℎ^2𝑖ℎ_2𝑖−1^𝑇,Λ^𝑖=⎛⎜⎜⎝ℎ^2𝑖ℎ^2𝑖−1⎞⎟⎟⎠^𝑇,Υ_𝑖=ℎ^2𝑖ℎ_2𝑖+1^𝑇.(2.8)

The SUT and the ISUT of {ℎ_𝑛}_𝑛∈ℤ by 𝐺 are, respectively, denoted by ℎ^𝑛_𝑛∈ℤ=Gℎ_𝑛_𝑛∈ℤ,ℎ^𝑛_𝑛∈ℤ=𝐆⁻ℎ_𝑛_𝑛∈ℤ.(2.9) By directly calculating, we have the following results.

Lemma 2.2. If {ℎ_𝑛}_𝑛∈ℤ is a one-dimensional CQF, then {ℎ_𝑛}_𝑛∈ℤ=𝐆{ℎ_𝑛}_𝑛∈ℤ and {ℎ_𝑛}_𝑛∈ℤ=𝐆⁻{ℎ_𝑛}_𝑛∈ℤ are one-dimensional CQFs.

Lemma 2.3. If {ℎ_𝑛}_𝑛∈ℤ=𝐆{ℎ_𝑛}_𝑛∈ℤ, then {ℎ_𝑛}_𝑛∈ℤ=(𝐆^𝑇)⁻{ℎ_𝑛}_𝑛∈ℤ.

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

Theorem 2.4. If {ℎ_𝑛}_𝑛∈ℤ 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 {ℎ_𝑛}_𝑛∈ℤ, we can get a simple one-dimensional CQF.
Without loss of generality, we assume that the support of {ℎ_𝑛}_𝑛∈ℤ is in {0,1,2,…, 2𝑁−1} and ℎ₀ℎ_2𝑁−1≠0. We will prove this theorem by induction.
We see that the theorem is true for 𝑁=1. Assume it is true for the case 𝑁≤𝐿(𝐿≥1,𝐿∈ℤ). We now prove it is true for the case 𝑁=𝐿+1. Suppose that {ℎ_𝑛}_𝑛∈ℤ is a one-dimensional CQF and ℎ₀ℎ_2𝐿+1≠0. We denote that 𝐺=−cosisn𝜃𝜃csoins𝜃𝜃,ℎ_𝑛_𝑛∈ℤ=𝐆ℎ_𝑛_𝑛∈ℤ,(2.10) where tan𝜃=−ℎ₁/ℎ₀. Recall that {ℎ_𝑛}_𝑛∈ℤ is a one-dimensional CQF, that is, ∑_𝑛∈ℤℎ_𝑛ℎ_𝑛+2𝑚=𝛿_0,𝑚, which implies that ℎ₀ℎ_2𝐿+ℎ₁ℎ_2𝐿+1=0. Therefore, ℎℎ₋₁=sin𝜃ℎ₀+cos𝜃ℎ₁=−cos𝜃ℎ₁+cos𝜃ℎ₁=0,_2𝐿=cos𝜃ℎ_2𝐿−sin𝜃ℎ_2𝐿+1=cos𝜃ℎ_2𝐿+cos𝜃ℎ¹ℎ₀ℎ_2𝐿+1=0.(2.11) It follows that the support of {ℎ_𝑛}_𝑛∈ℤ is in {0,1,2,…,2𝐿−1}. Suppose that −𝜋/2<𝜃<𝜋/2, then ℎ₀≠0. It follows that there exists 𝑡∈ℤ(𝑡≤𝐿), such that ℎ₀ℎ_2𝑡−1≠0. Furthermore, if 𝑛∈ℤ⧵{0,1,2,…,2𝑡−1}, then ℎ_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 {ℎ_𝑛}_𝑛∈ℤ is a one-dimensional CQF. By Theorem 2.4, we know that {ℎ_𝑛}_𝑛∈ℤ can be constructed by a simple one-dimensional CQF and a series of 2×2 orthogonal matrices. For a real number 𝜃∈ℝ, we denote 𝐺_𝜃 as the 2×2 orthogonal matrix: −cosisn𝜃𝜃csoins𝜃𝜃,𝜃∈ℝ.(2.12) Let {ℋ_𝑛}^{𝛾𝑛∈ℤ} be a simple one-dimensional CQF such that ℋ_𝑛=⎧⎪⎪⎨⎪⎪⎩cs0io,ns𝛾𝛾,,𝑛𝑛ot==her01w,,ise.𝛾∈ℝ.(2.13) For an arbitrary positive integer 𝑁, choosing 𝛾₀,𝛾₁,…,𝛾_𝑁−1∈ℝ, we define {ℎ^𝑁𝑛}_𝑛∈ℤ as follows: ℎ^𝑁𝑛_𝑛∈ℤ=𝐆_𝛾𝑁−1𝐆_𝛾𝑁−2⋯𝐆_𝛾1ℋ_𝑛^𝛾0_𝑛∈ℤ.(2.14) In general, the support of {ℎ^𝑁𝑛}_𝑛∈ℤ is in [0,2𝑁−1]∩ℤ. In other words, {ℎ^𝑁𝑛}_𝑛∈ℤ 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 𝑁−1 times SUT of one-dimensional CQF.

We can prove inductively the following theorems.

Theorem 2.5. Let 𝜂=𝛾₀+𝛾₁+⋯+𝛾_𝑁−1, then _𝑖∈ℤℎ^𝑁2𝑖=cos𝜂,_𝑖∈ℤℎ^𝑁2𝑖+1=sin𝜂.(2.15)

Theorem 2.6. Let 𝛾₀+𝛾₁+⋯+𝛾_𝑁−1=2𝑘𝜋+𝜋4,𝑘∈ℤ,(2.16) then the sequence {ℎ^𝑁𝑛}_𝑛∈ℤ given in (2.14) is a one-dimensional low-pass wavelet filter.

Theorem 2.6 provides a condition for choosing 𝛾₀,𝛾₁,…,𝛾_𝑁−1 such that the one-dimensional CQF {ℎ^𝑁𝑛}_𝑛∈ℤ given in (2.14) is a one-dimensional orthogonal low-pass wavelet filter.

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

By Theorem 2.6, we know that a length-2𝑁 one-dimensional low-pass wavelet filter {ℎ^𝑁𝑛}_𝑛∈ℤ can be constructed by choosing 𝛾₀,𝛾₁,…,𝛾_𝑁−1 such that condition (2.16) is satisfied. Therefore, any length-2𝑁 one-dimensional low-pass wavelet filter can be parameterized into a (𝑁−1)-parameter family of wavelet system. In fact, we can give an explicit parametrization of any length-2𝑁 filter: ℎ^𝑁𝑛_𝑛∈ℤ=𝐆_𝛾𝑁−1𝐆_𝛾𝑁−2⋯𝐆_𝛾1ℋ_𝑛^{𝜋/4−𝛾𝑁−1−𝛾𝑁−2−⋯−𝛾1}_𝑛∈ℤ,(2.17) where 𝛾₁,𝛾₂,…,𝛾_𝑁−1∈ℝ.

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, 24–26]. 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)=2_𝛼∈ℤ2𝑏_𝛼𝜙(2x−𝛼)(3.1) and its associated wavelets 𝜓^𝑙(x)=2_𝛼∈ℤ2𝑑^𝑙𝛼𝜙(2x−𝛼),𝑙=1,2,3,(3.2) we need construct sequences {𝑏_𝛼}_𝛼∈ℤ² and {𝑑^𝑙𝛼}_𝛼∈ℤ² such that (see [10, 12, 13], etc.) _𝛼_∈ℤ2𝑏_𝛼=2,(3.3)_𝛼∈ℤ2𝑏_𝛼𝑏_𝛼+2𝛽=𝛿_0𝛽,(3.4)_{𝑖,𝑗∈ℤ}𝑏_{(2𝑖,2𝑗)}=_{𝑖,𝑗∈ℤ}𝑏_{(2𝑖+1,2𝑗)}=_{𝑖,𝑗∈ℤ}𝑏_{(2𝑖,2𝑗+1)}=_{𝑖,𝑗∈ℤ}𝑏_{(2𝑖+1,2𝑗+1)},(3.5)_𝛼∈ℤ2𝑑^𝑗1_𝛼𝑑^𝑗2_𝛼+2𝛽=𝛿_𝑗1,𝑗2𝛿_0,𝛽,(3.6)_𝛼∈ℤ2𝑏_𝛼𝑑^{𝑗𝛼+2𝛽}=0,(3.7) where 𝑗,𝑗₁,𝑗₂=1,2,3 and 𝛽∈ℤ².

The sequence {𝑏_𝛼}_𝛼∈ℤ², which satisfies (3.4), is called a two-dimensional CQF. If {𝑏_𝛼}_𝛼∈ℤ² satisfies (3.3), (3.4), and (3.5) simultaneously, we call it a two-dimensional low-pass wavelet filter. The sequences {𝑑^𝑗𝛼}_𝛼∈ℤ²(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 {𝑙_𝛼}_𝛼∈ℤ²∈ℓ²(ℤ²), we define Δ=𝛼∶𝛼∈ℤ²,𝑙_𝛼≠0,(3.8) and Δ is called the support set of {𝑙_𝛼}_𝛼∈ℤ²∈ℓ²(ℤ²). If Δ is finite, we call {𝑙_𝛼}_𝛼∈ℤ²∈ℓ²(ℤ²) finite supported sequence. As aforementioned, the sequences we consider are real-valued and finite supported (FIR).

We note that any sequence {𝑏_𝛼}_𝛼∈ℤ² can be split into 4 disjoint subsets 𝑏𝑏_{(2𝑖,2𝑗)}∶𝑖,𝑗∈ℤ,𝑏_{(2𝑖,2𝑗+1)}∶𝑖,𝑗∈ℤ_{(2𝑖+1,2𝑗)},∶𝑖,𝑗∈ℤ,𝑏_{(2𝑖+1,2𝑗+1)}∶𝑖,𝑗∈ℤ.(3.9)

Definition 3.1. Let 𝑈 be an arbitrary 4×4 orthogonal matrix, and let {𝑏_𝛼}_𝛼∈ℤ² be an FIR. For integers𝐴_𝑠,𝐵_𝑠(𝑠=1,2,3,4) and for all 𝑖,𝑗∈ℤ, we set Γ^𝑖,𝑗=⎛⎜⎜⎜⎜⎜⎝𝑏̃^{(2𝑖,2𝑗)}𝑏̃_{(2𝑖,2𝑗+1)}𝑏̃_{(2𝑖+1,2𝑗)}𝑏̃_{(2𝑖+1,2𝑗+1)})⎞⎟⎟⎟⎟⎟⎠^𝑇Γ_{𝑖,𝑗(𝐴1,𝐵1,𝐴2,𝐵2,𝐴3,𝐵3,𝐴4,𝐵4)},=⎛⎜⎜⎜⎜⎜⎝𝑏^{(2𝑖−2𝐴1,2𝑗−2𝐵1)}𝑏_{(2𝑖−2𝐴2,2𝑗−2𝐵2+1)}𝑏_{(2𝑖−2𝐴3+1,2𝑗−2𝐵3)}𝑏_{(2𝑖−2𝐴4+1,2𝑗−2𝐵4+1)}⎞⎟⎟⎟⎟⎟⎠^𝑇.(3.10) Then {𝑏̃_𝛼}_𝛼∈ℤ², which is defined as follows: Γ^𝑖,𝑗=Γ^{𝑖,𝑗(𝐴}_{1,𝐵1,𝐴2,𝐵2,𝐴3,𝐵3,𝐴4,𝐵4)}𝑈(3.11) is called the two-dimensional SUT of {𝑏_𝛼}_𝛼∈ℤ².

Lemma 3.2. If {𝑏_𝛼}_𝛼∈ℤ² is a two-dimensional CQF and {𝑏̃_𝛼}_𝛼∈ℤ² is given by (3.11), then {𝑏̃_𝛼}_𝛼∈ℤ² is also a two-dimensional CQF.

Proof. By directly calculating, we can prove that {𝑏̃_𝛼}_𝛼∈ℤ² satisfies the following equation: _𝛼_∈ℤ2𝑏̃_𝛼𝑏̃_𝛼+2𝛽=𝛿_0,𝛽,∀𝛽∈ℤ².(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 {𝑏_𝛼}_𝛼∈ℤ² is a two-dimensional CQF, {𝑑^𝑘𝛼}_𝛼∈ℤ²(𝑘=1,2,3) satisfy (3.6) and (3.7), {𝑏̃_𝛼}_𝛼∈ℤ² is a two-dimensional CQF obtained by SUT of {𝑏_𝛼}_𝛼∈ℤ², then {𝑑^𝑘𝛼}_𝛼∈ℤ², which are derived by SUT of {𝑑^𝑘𝛼}_𝛼∈ℤ²(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}̃_𝛼∈ℤ² is called the SUT0 of {𝑏_𝛼}_𝛼∈ℤ² if one chooses 𝐴_𝑖=𝐵_𝑖=0(𝑖=1,2,3,4)(3.13) in (3.11). This transform is denoted by {𝑏̃_𝛼}_𝛼∈ℤ²=⁰𝐔{𝑏_𝛼}_𝛼∈ℤ².
(ii) {𝑏}̃_𝛼∈ℤ² is called the SUTT1 of {𝑏_𝛼}_𝛼∈ℤ² if one chooses 𝐵₂=𝐵₄=1,𝐵₁=𝐵₃=𝐴_𝑖=0(𝑖=1,2,3,4)(3.14) in (3.11). This transform is denoted by {𝑏̃_𝛼}_𝛼∈ℤ²=^𝑇1𝐔{𝑏_𝛼}_𝛼∈ℤ².{𝑏}̃_𝛼∈ℤ² is called the SUTT2 of {𝑏_𝛼}_𝛼∈ℤ² if one chooses 𝐴₃=𝐴₄=1,𝐴₁=𝐴₂=𝐵_𝑖=0(𝑖=1,2,3,4).(3.15) This transform is denoted by {𝑏̃_𝛼}_𝛼∈ℤ²=^𝑇2𝐔{𝑏_𝛼}_𝛼∈ℤ².
(iii) {𝑏}̃_𝛼∈ℤ² is called the SUT1 of {𝑏_𝛼}_𝛼∈ℤ² if one chooses 𝐴₂=𝐴₄=1,𝐴₁=𝐴₃=𝐵_𝑖=0(𝑖=1,2,3,4)(3.16) in (3.11). This transform is denoted by {𝑏̃_𝛼}_𝛼∈ℤ²=¹𝐔{𝑏_𝛼}_𝛼∈ℤ².{𝑏}̃_𝛼∈ℤ² is called the SUT2 of {𝑏_𝛼}_𝛼∈ℤ² if we choose 𝐵₃=𝐵₄=1,𝐵₁=𝐵₂=𝐴_𝑖=0(𝑖=1,2,3,4).(3.17) This transform is denoted by {𝑏̃_𝛼}_𝛼∈ℤ²=²𝐔{𝑏_𝛼}_𝛼∈ℤ².

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 𝑏_𝛼=⎧⎪⎪⎨⎪⎪⎩14√if𝛼=(0,0)or(1,1),304oifth𝛼er=w(is1e,0)or(0,1),(3.18) is {(0,0),(0,1),(1,0),(1,1)}, and the support of the filter {𝑏̃_𝛼}_𝛼∈ℤ2=¹𝐔{𝑏_𝛼}_𝛼∈ℤ² is {(0,0),(0,1),(1,0),(1,1),(2,0),(2,1),(3,0),(3,1)}⊂([0,3]×[0,1]∩ℤ²), where 𝑈=diag(𝐻,𝐻) and 𝐻=⎡⎢⎢⎢⎢⎣12√3−√23212⎤⎥⎥⎥⎥⎦.(3.19) In general, for integers 𝑁,𝑀, if the support of {𝑏_𝛼}_𝛼∈ℤ² is in [0,2𝑁−1]×[0,2𝑀−1]∩ℤ², after SUT1 (or SUT2) the support of the new filter is in [0,2𝑁+1]×[0,2𝑀−1]∩ℤ² (or in [0,2𝑁−1]×[0,2𝑀+1]∩ℤ²). If the support of {𝑏_𝛼}_𝛼∈ℤ² is {(0,0),(1,0),(0,1),(1,1)}, we call {𝑏_𝛼}_𝛼∈ℤ² a simple two-dimensional filter.

We will adopt the following notations in the rest of this paper. For arbitrary 𝜉₀,𝜆₀,𝜉,𝜆∈𝑅, let {ℬ_𝛼}^𝜆0,𝜉0_𝛼∈ℤ2 be the FIR defined as follows: ℬ_𝛼=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩cos𝜉⁰cos𝜉₀cos𝜆₀,𝛼=(0,0);sin𝜉₀sin𝜆₀,𝛼=(0,1);sin𝜉₀cos𝜆₀,𝛼=(1,0);0,sin𝜆₀,𝛼oth=er(w1,i1se).;(3.20) Furthermore, let {𝒟^𝑘𝛼}^𝜆0,𝜉0_𝛼∈ℤ2(𝑘=1,2,3) be the filters as follows 𝒟^1𝛼=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩cos𝜉⁰−cos𝜉s₀in𝜆₀,𝛼=(0,0);sin𝜉₀cos𝜆₀,𝛼=(0,1);−sin𝜉s₀in𝜆₀,𝛼=(1,0);𝒟0,cos𝜆₀,𝛼oth=er(w1,i1se),_2𝛼;=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩sin𝜉⁰sin𝜉₀cos𝜆₀,𝛼=(0,0);−cos𝜉si₀n𝜆₀,𝛼=(0,1);−cos𝜉₀cos𝜆₀,𝛼=(1,0);𝒟0,sin𝜆₀,𝛼ot=he(r1w,1is)e;,(3.21)_3𝛼=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩sin𝜉⁰−sin𝜉s₀in𝜆₀,𝛼=(0,0);−cos𝜉₀cos𝜆₀,𝛼=(0,1);cos𝜉₀sin𝜆₀,𝛼=(1,0);0,cos𝜆₀,𝛼ot=he(r1w,1is)e;,(3.22) and let 𝑈_𝜆,𝑈_𝜉 be orthogonal matrices such that 𝑈𝑈_𝜆=⎛⎜⎜⎜⎜⎜⎝−cos00isn𝜆𝜆csoin00s𝜆𝜆−cos00isn𝜆𝜆csoin00s𝜆𝜆_𝜉⎞⎟⎟⎟⎟⎟⎠,=⎛⎜⎜⎜⎜⎜⎝−cos00isn𝜉𝜉−cos00isn𝜉𝜉csoi00ns𝜉𝜉csoi00ns𝜉𝜉⎞⎟⎟⎟⎟⎟⎠,(3.23) respectively. Then {ℬ_𝛼}^𝜆0,𝜉0_𝛼∈ℤ2 is a simple two-dimensional CQF, and {𝒟^𝑘𝛼}^𝜆0,𝜉0_𝛼∈ℤ2 satisfy (3.6) and (3.7).

From the special two-dimensional CQF {ℬ_𝛼}^𝜆0,𝜉0_𝛼∈ℤ2, by SUTT1, SUTT2, SUT1, SUT2, and the matrices 𝑈_𝜆,𝑈_𝜉, 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 {𝑏_𝛼}_𝛼∈ℤ² is called tensor-product wavelet filter if it satisfies the following equations: 𝑏𝑏_{(2𝑖,2𝑗)(2𝑖,2𝑗+1)}=ℎ_2𝑖ℎ_2𝑗,𝑏_{(2𝑖+1,2𝑗)}=ℎ_2𝑖ℎ_2𝑗+1,𝑏_{(2𝑖+1,2𝑗+1)}=ℎ_2𝑖+1ℎ_2𝑗,=ℎ_2𝑖+1ℎ_2𝑗+1,𝑖,𝑗∈ℤ,(3.24) where {ℎ_𝑖}_𝑖∈ℤ, {ℎ_𝑖}_𝑖∈ℤ are one-dimensional orthogonal low-pass wavelet filters.

Theorem 3.5. If {𝑏_𝛼}_𝛼∈ℤ² is a tensor-product low-pass wavelet filter, then, it can be constructed by SUTT1 and SUTT2 from {ℬ_𝛼}^𝜆0,𝜉0_𝛼∈ℤ2.

Proof. Forall𝑖,𝑗∈ℤ, it follows from (3.24) that 𝑏_{(2𝑖,2𝑗)},𝑏_{(2𝑖,2𝑗+1)},𝑏_{(2𝑖+1,2𝑗)},𝑏_{(2𝑖+1,2𝑗+1)}=ℎ_2𝑖,ℎ_2𝑖,ℎ_2𝑖+1,ℎ_2𝑖+1⎛⎜⎜⎜⎜⎜⎜⎝ℎ^2𝑗0ℎ000_2𝑗+100ℎ0_2𝑗0000ℎ0_2𝑗+1⎞⎟⎟⎟⎟⎟⎟⎠.(3.25) Suppose that {ℎ_𝑖}_𝑖∈ℤ is a length-2𝑁 one-dimensional filter. Then there exist 𝜆₀,…,𝜆_𝑁−2,𝜆_𝑁−1∈𝑅 such that 𝜆₀+⋯+𝜆_𝑁−2+𝜆_𝑁−1=𝜋/4, and ℎ^𝑛_𝑛∈ℤ=𝐆_𝜆𝑁−1𝐆_𝜆𝑁−2⋯𝐆_𝜆1ℋ_𝑛^𝜆0_𝑛∈ℤ.(3.26) We denote ℎ^𝑛^{𝑁−1𝑛∈ℤ}=𝐆_𝜆𝑁−2⋯𝐆_𝜆1ℋ_𝑛^𝜆0_𝑛∈ℤ.(3.27) Let {𝑏_𝛼}^{𝑁−1𝛼∈ℤ}₂ be two-dimensional filter such that 𝑏𝑏^{𝑁−1(2𝑖,2𝑗)}_{𝑁−1(2𝑖,2𝑗+1)}=ℎ_2𝑖ℎ^{𝑁−12𝑗},𝑏_{𝑁−1(2𝑖+1,2𝑗)}=ℎ_2𝑖ℎ^{𝑁−12𝑗+1},𝑏_{𝑁−1(2𝑖+1,2𝑗+1)}=ℎ_2𝑖+1ℎ^{𝑁−12𝑗},=ℎ_2𝑖+1ℎ^{𝑁−12𝑗+1},𝑖,𝑗∈ℤ.(3.28) It follows that 𝑏=_{(2𝑖,2𝑗)},ℎ𝑏_{(2𝑖,2𝑗+1)2𝑖},𝑏_{(2𝑖+1,2𝑗)},𝑏_{(2𝑖+1,2𝑗+1)},ℎ_2𝑖,ℎ_2𝑖+1,ℎ_2𝑖+1⎛⎜⎜⎜⎜⎜⎜⎝ℎ^2𝑗0ℎ000_2𝑗+100ℎ0_2𝑗0000ℎ0_2𝑗+1=ℎ_2𝑖⎞⎟⎟⎟⎟⎟⎟⎠,ℎ_2𝑖,ℎ_2𝑖+1,ℎ_2𝑖+1⎛⎜⎜⎜⎜⎜⎜⎜⎝ℎ^{𝑁−12𝑗}0ℎ000_{𝑁−12𝑗−1}00ℎ0_{𝑁−12𝑗}0000ℎ0_{𝑁−12𝑗−1}×⎛⎜⎜⎜⎜⎜⎝cos𝜆_𝑁−1⎞⎟⎟⎟⎟⎟⎟⎟⎠−sin𝜆_𝑁−1sin𝜆_𝑁−1000cos𝜆0_𝑁−1cos𝜆0_𝑁−10sin𝜆_𝑁−100−sin𝜆_𝑁−1cos𝜆_𝑁−1=𝑏_{𝑁−1(2𝑖,2𝑗)}⎞⎟⎟⎟⎟⎟⎠,𝑏^{𝑁−1(2𝑖,2𝑗−1)},𝑏^{𝑁−1(2𝑖+1,2𝑗)},𝑏^{𝑁−1(2𝑖+1,2𝑗−1)}𝑈^𝜆_𝑁−1^.(3.29) Hence, {𝑏_𝛼}_𝛼∈ℤ² can be constructed by SUTT1 of {𝑏_𝛼}^{𝑁−1𝛼∈ℤ}₂.
It can be proved inductively that {𝑏_𝛼}_𝛼∈ℤ² can be constructed by a series of SUTT1 from {𝑏_𝛼}^0𝛼∈ℤ₂, and {𝑏_𝛼}^0𝛼∈ℤ₂ can be constructed by a series of SUTT2 from {ℬ_𝛼}^𝜆0,𝜉0_𝛼∈ℤ2, where {𝑏_𝛼}^0𝛼∈ℤ₂ denotes the filter 𝑏𝑏^{0(2𝑖,2𝑗)}_{0(2𝑖,2𝑗+1)}=ℎ_2𝑖ℋ_2𝑗,𝑏_{0(2𝑖+1,2𝑗)}=ℎ_2𝑖ℋ_2𝑗+1,𝑏_{0(2𝑖+1,2𝑗+1)}=ℎ_2𝑖+1ℋ_2𝑗,=ℎ_2𝑖+1ℋ_2𝑗+1,𝑖,𝑗∈ℤ.(3.30) Therefore, any tensor-product two-dimensional low-pass wavelet filters can be constructed by SUTT1 and SUTT2 from {ℬ_𝛼}^𝜆0,𝜉0_𝛼∈ℤ2.

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,…,𝜆_𝑁−1,𝜉₀,𝜉₁,…,𝜉_𝑀−1∈ℝ, {𝑏^{𝛼𝑁,𝑀}}_𝛼∈ℤ² is defined as 𝑏^{𝛼𝑁,𝑀}_𝛼∈ℤ²=^{𝜀𝑛𝑁+𝑀−2}𝑈_{𝑛𝑁+𝑀−2}^{𝜀𝑛𝑁+𝑀−3}𝑈_{𝑛𝑁+𝑀−3}⋯^𝜀𝑛1𝑈_𝑛1ℬ_𝛼^𝜆0,𝜉0_𝛼∈ℤ²,(3.31) where 𝑛₁,𝑛₂,…,𝑛_{𝑁+𝑀−2}∈{𝜆₁,𝜆₂,…,𝜆_𝑁−1,𝜉₁,𝜉₂,…,𝜉_𝑀−1} and 𝜀_𝑛𝑗=⎧⎪⎨⎪⎩1,if𝑛^𝑗2,if𝑛_𝑗∈𝜆₁,𝜆₂,…,𝜆_𝑁−1,∈𝜉₁,𝜉₂,…,𝜉_𝑀−1.(3.32) Then {𝑏^{𝛼𝑁,𝑀}}_𝛼∈ℤ² is a two-dimensional CQF, and its support is in [0,2𝑁−1]×[0,2𝑀−1]∩ℤ². We are now in a position to draw some conditions on choosing 𝜆₀,𝜆₁,…,𝜆_𝑁1,𝜉₀,𝜉₁,…,𝜉_𝑀1, such that {𝑏^{𝛼𝑁,𝑀}}_𝛼∈ℤ² is a two-dimensional low-pass wavelet filter.

Theorem 3.6. Let 𝜂=𝜆₀+𝜆₁+⋯+𝜆_𝑁−1,𝜂̃=𝜉₀+𝜉₁+⋯+𝜉_𝑀−1.Then _𝑖_,𝑗∈ℤ𝑏^{𝑁,𝑀(2𝑖,2𝑗)}=cos𝜂cos𝜂̃,_{𝑖,𝑗∈ℤ}𝑏^{𝑁,𝑀(2𝑖,2𝑗+1)}=sin𝜂cos𝜂̃,(3.33)_{𝑖,𝑗∈ℤ}𝑏^{𝑁,𝑀(2𝑖+1,2𝑗)}=cos𝜂sin𝜂̃,_{𝑖,𝑗∈ℤ}𝑏^{𝑁,𝑀(2𝑖+1,2𝑗+1)}=sin𝜂sin𝜂̃.(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 𝑁≤𝑘₁,𝑀≤𝑘₂(𝑘₁≥1,𝑘₂≥1,𝑘₁,𝑘₂∈ℤ); now we prove that it is true for the case 𝑁=𝑘₁+1,𝑀=𝑘₂.
Let 𝑠 be the integer such that 𝑛_𝑠+1∈{𝜆₁,𝜆₂,…,𝜆_𝑘1} and 𝑛_𝑠+𝑡∈{𝜉₁,𝜉₂,…,𝜉_𝑘2}(𝑡>1,𝑡∈ℤ). Suppose that 𝑖𝑗₁₁,𝑖₂,…,𝑖_𝑢≤𝑠⟹𝑛_𝑖1,𝑛_𝑖2,…,𝑛_𝑖𝑢∈𝜆₁,𝜆₂,…,𝜆_𝑘1,,𝑗₂,…,𝑗_𝑣≤𝑠⟹𝑛_𝑗1,𝑛_𝑗2,…,𝑛_𝑗𝑣∈𝜉₁,𝜉₂,…,𝜉_𝑘1.(3.35) It follows that 𝑢=𝑘₁ and 𝑣≤𝑘₂−1. Let 𝜂_𝑠=𝜆₀+𝑏𝑛_{𝑖1𝑠𝛼}+𝑛_𝑖2+⋯+𝑛_𝑖𝑢,𝜂̃_𝑠=𝜉₀+𝑛_𝑗1+𝑛_𝑗2+⋯+𝑛_𝑗𝑣,_𝛼∈ℤ²=^𝜀𝑛𝑠𝑈_𝑛𝑠^{𝜀𝑛𝑠−1}𝑈_𝑛𝑠−1⋯^𝜀𝑛1𝑈_𝑛1ℬ_𝛼^𝜆0,𝜉0_𝛼∈ℤ²;(3.36) it follows that _𝑖_{,𝑗∈ℤ𝑖}_,𝑗∈ℤ𝑏^{𝑠(2𝑖,2𝑗)}=cos𝜂_𝑠cos𝜂̃_𝑠,_{𝑖,𝑗∈ℤ}𝑏^{𝑠(2𝑖,2𝑗+1)}=sin𝜂_𝑠cos𝜂̃_𝑠,𝑏^{𝑠(2𝑖+1,2𝑗)}=cos𝜂_𝑠sin𝜂̃_𝑠,_{𝑖,𝑗∈ℤ}𝑏^{𝑠(2𝑖+1,2𝑗+1)}=sin𝜂_𝑠sin𝜂̃_𝑠.(3.37) Let {𝑏^𝛼𝑠+1}_𝛼∈ℤ2=^{𝜀𝑛𝑠+1}𝑈_𝑛𝑠+1{𝑏^𝑠𝛼}_𝛼∈ℤ2. Since 𝑛_𝑠+1∈{𝜆₁,𝜆₂,…,𝜆_𝑘1}, then 𝑏𝑏^{𝑠+1(2𝑖,2𝑗)}_{𝑠+1(2𝑖,2𝑗+1)}=cos𝑛_𝑠+1𝑏^{𝑠(2𝑖,2𝑗)}−sin𝑛_𝑠+1𝑏^{𝑠(2𝑖−2,2𝑗+1)},𝑏_{𝑠+1(2𝑖+1,2𝑗)}=sin𝑛_𝑠+1𝑏^{𝑠(2𝑖,2𝑗)}+cos𝑛_𝑠+1𝑏^{𝑠(2𝑖−2,2𝑗+1)},𝑏_{𝑠+1(2𝑖+1,2𝑗+1)}=cos𝑛_𝑠+1𝑏^{𝑠(2𝑖+1,2𝑗)}−sin𝑛_𝑠+1𝑏^{𝑠(2𝑖−1,2𝑗+1)},=sin𝑛_𝑠+1𝑏^{𝑠(2𝑖+1,2𝑗)}+cos𝑛_𝑠+1𝑏^{𝑠(2𝑖−1,2𝑗+1)}.(3.38) Therefore, _𝑖_{,𝑗∈ℤ𝑖}_,𝑗∈ℤ𝑏^{𝑠+1(2𝑖,2𝑗)}=cos𝜂_𝑠+𝑛_𝑠+1cos𝜂̃_𝑠;_{𝑖,𝑗∈ℤ}𝑏^{𝑠+1(2𝑖,2𝑗+1)}=sin𝜂_𝑠+𝑛_𝑠+1cos𝜂̃_𝑠;𝑏^{𝑠+1(2𝑖+1,2𝑗)}=cos𝜂_𝑠+𝑛_𝑠+1sin𝜂̃_𝑠;_{𝑖,𝑗∈ℤ}𝑏^{𝑠+1(2𝑖+1,2𝑗+1)}=sin𝜂_𝑠+𝑛_𝑠+1sin𝜂̃_𝑠.(3.39) For 𝑡>1, let 𝑏^{𝛼𝑠+𝑡}_𝛼∈ℤ²=^{𝜀𝑛𝑠+𝑡}𝑈_{𝑛𝑠+𝑡}⋯^{𝜀𝑛𝑠+1}𝑈_𝑛𝑠+1𝑏^𝑠𝛼_𝛼∈ℤ².(3.40) It follows from 𝑛_𝑠+𝑡∈{𝜉₁,𝜉₂,…,𝜉_𝑘2−1} that 𝑏𝑏^{𝑠+𝑡(2𝑖,2𝑗)}_{𝑠+𝑡(2𝑖,2𝑗+1)}=cos𝑛_𝑠+𝑡𝑏^{𝑠+𝑡−1(2𝑖,2𝑗)}−sin𝑛_𝑠+𝑡𝑏^{𝑠+𝑡−1(2𝑖+1,2𝑗−2)},𝑏_{𝑠+𝑡(2𝑖+1,2𝑗)}=cos𝑛_𝑠+𝑡𝑏^{𝑠+𝑡−1(2𝑖,2𝑗+1)}−sin𝑛_𝑠+𝑡𝑏^{𝑠+𝑡−1(2𝑖+1,2𝑗−1)},𝑏_{𝑠+𝑡(2𝑖+1,2𝑗+1)}=sin𝑛_𝑠+𝑡𝑏^{𝑠+𝑡−1(2𝑖,2𝑗)}+cos𝑛_𝑠+𝑡𝑏^{𝑠+𝑡−1(2𝑖+1,2𝑗−2)},=sin𝑛_𝑠+𝑡𝑏^{𝑠+𝑡−1(2𝑖,2𝑗+1)}+cos𝑛_𝑠+𝑡𝑏^{𝑠+𝑡−1(2𝑖+1,2𝑗−1)}.(3.41) Therefore, _𝑖_{,𝑗∈ℤ𝑖}_,𝑗∈ℤ𝑏^{𝑠+𝑡(2𝑖,2𝑗)}=cos𝜂_{𝑠+𝑡−1}cos𝜂̃_{𝑠+𝑡−1}+𝑛_𝑠+𝑡;_𝑖_,𝑗∈ℤ𝑏^{𝑠+𝑡(2𝑖,2𝑗+1)}=sin𝜂_{𝑠+𝑡−1}cos𝜂̃_{𝑠+𝑡−1}+𝑛_𝑠+𝑡;_𝑖_,𝑗∈ℤ𝑏^{𝑠+𝑡(2𝑖+1,2𝑗)}=cos𝜂_{𝑠+𝑡−1}sin𝜂̃_{𝑠+𝑡−1}+𝑛_𝑠+𝑡;𝑏^{𝑠+𝑡(2𝑖+1,2𝑗+1)}=sin𝜂_{𝑠+𝑡−1}sin𝜂̃_{𝑠+𝑡−1}+𝑛_𝑠+𝑡,(3.42) where 𝜂_{𝑠+𝑡−1}=𝜆₀+𝜆₁+⋯+𝜆_𝑘1,𝜂̃_{𝑠+𝑡−1}=𝜉₀+𝑛_𝑗1+⋯+𝑛_𝑗𝑣+𝑛_𝑠+2+𝑛_{𝑠+𝑡−1}.(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,…,𝜆_𝑁1,𝜉₀,𝜉₁,…,𝜉_𝑀1, such that {𝑏^{𝛼𝑁,𝑀}}_𝛼∈ℤ² is a two-dimensional low-pass wavelet filter.

Theorem 3.7. If there exists integers 𝑛₁ and 𝑛₂, such that 𝜂=𝜆₀+𝜆₁+⋯+𝜆_𝑁−1=2𝑛₁𝜋+𝜋𝜂̃=𝜉₀4,(3.44)+𝜉₁+⋯+𝜉_𝑀−1=2𝑛₂𝜋+𝜋4,(3.45) then the sequence {𝑏^{𝛼𝑁,𝑀}}_𝛼∈ℤ² constructed in (3.31) is a two-dimensional low-pass wavelet filter.

Proof. It follows from Theorem 3.6 that _𝑖_,𝑗∈ℤ𝑏_{(2𝑖,2𝑗)}=cos𝜂cos𝜂̃=12,_{𝑖,𝑗∈ℤ}𝑏_{(2𝑖,2𝑗+1)}=sin𝜂cos𝜂̃=1_𝑖_,𝑗∈ℤ2,𝑏_{(2𝑖+1,2𝑗)}=cos𝜂sin𝜂̃=12,_{𝑖,𝑗∈ℤ}𝑏_{(2𝑖+1,2𝑗+1)}=sin𝜂sin𝜂̃=12.(3.46) Therefore, {𝑏^{𝛼𝑁,𝑀}}_𝛼∈ℤ² satisfies conditions (3.3) and (3.5), then it is a two-dimensional low-pass wavelet filter.

Corollary 3.8. If {𝑏^{𝛼𝑁,𝑀}}_𝛼∈ℤ² as given in (3.31) is a low-pass wavelet filter, then 𝑑^{𝛼𝑘,𝑁,𝑀}_𝛼∈ℤ²=^{𝜀𝑛𝑁+𝑀−2}𝑈_{𝑛𝑁+𝑀−2}^{𝜀𝑛𝑁+𝑀−3}𝑈_{𝑛𝑁+𝑀−3}⋯^𝜀𝑛1𝑈_𝑛1𝒟^𝑘𝛼^𝜆0,𝜉0_𝛼∈ℤ²(3.47) are the high-pass filters associated with {𝑏^{𝛼𝑁,𝑀}}_𝛼∈ℤ², where 𝑘=1,2,3.

Remark 3.9. We can choose other orthogonal matrices than U_𝜆 and U_𝜉 such as 𝑈𝑈_𝜆,𝜆̃=⎛⎜⎜⎜⎜⎜⎝−cos00isn𝜆𝜆csoin00s𝜆𝜆−cos00isn𝜆𝜆̃̃csoin00s_𝜉,𝜆𝜆̃̃⎞⎟⎟⎟⎟⎟⎠,𝜉̃=⎛⎜⎜⎜⎜⎜⎝−cos00isn𝜉𝜉−cos00isn𝜉𝜉̃̃csoi00ns𝜉𝜉csoi00ns𝜉𝜉̃̃⎞⎟⎟⎟⎟⎟⎠,(3.48) where 𝜆,𝜆̃,𝜉,𝜉̃∈ℝ; 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∈ℝ, define {𝑏_𝛼}_𝛼∈ℤ2 as follows: 𝑏_𝛼_𝛼∈ℤ²=²𝑈_𝜉1¹𝑈_𝜆1ℬ_𝛼^𝜆0,𝜉0_𝛼∈ℤ².(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/4−𝜉₁, 𝜆₁=4.357946, 𝜆₀=𝜋/4−𝜆₁. 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.

Table 1 

The coefficients of a nonseparable orthogonal low-pass wavelet filter.

Table 2 

The coefficients of a high-pass wavelet filter (𝑑^1(𝑖,𝑗)) associated with the low-pass wavelet filter in Table 1.

Table 3 

The coefficients of a high-pass wavelet filter (𝑑^2(𝑖,𝑗)) associated with the low-pass wavelet filter in Table 1.

Table 4 

The coefficients of a high-pass wavelet filter (𝑑^3(𝑖,𝑗)) associated with the low-pass wavelet filter in Table 1.

(a)

(b)

Figure 1 

It shows that the high-frequency subbands by the derived filter can reveal more features than that by the commonly used tensor-product wavelet filter. (a) Decomposition of the “Lena” image by the derived filter. (b) Decomposition of the “Lena” image by tensor-product filter. All the coefficients in the high-frequency subbands are magnified by a factor 20 to see the difference.

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.