# Shift Unitary Transform for Constructing Two-Dimensional Wavelet Filters

Authors: Fei Li; Jianwei Yang
Journal: Journal of Applied Mathematics
 (2011)

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## 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 asShift 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.


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## Body

## 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 asShift 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 matrixA, 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 Section2. 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(2.1)φ(x)=2∑k∈Zhkφ(2x-k),
we need construct sequences {hk}k∈ℤ such that (see [4, 21] and many others) (2.2)∑k∈Zhk=2,(2.3)∑k∈Zh2k=∑k∈Zh2k+1,(2.4)∑k∈Zhkhk+2α=δ0,α,α∈Z,
where δ denotes the Kronecker delta
(2.5)δα,β={0,α≠β,1,α=β.α,β∈Z.
The sequence {hk}k∈ℤ which satisfies (2.4) is called a one-dimensional CQF, and the sequence {hk}k∈ℤ, which satisfies (2.2), (2.3), (2.4) simultaneously, is called one-dimensional orthogonal low-pass wavelet filter.Definition 2.1.
Let{hn}n∈ℤ be a one-dimensional CQF and G an arbitrary 2×2 orthogonal matrix. {h̃n}n∈ℤ is called the  shift unitary transform (SUT) of {hn}n∈ℤby G if {h̃n}n∈ℤ satisfies
(2.6)Υ̃i=ΛiG,i∈Z,
and {hn′̃}n∈ℤ is called the  inverse shift unitary transform (ISUT) of {hn}n∈ℤ by G if {hn′̃}n∈ℤ satisfies
(2.7)Λi′̃=ΥiG,i∈Z,
where for any i∈Z,
(2.8)Υ̃i=(h̃2ih̃2i+1)T,Λi=(h2ih2i-1)T,Λi′̃=(h2i′̃h2i-1′̃)T,Υi=(h2ih2i+1)T.The SUT and the ISUT of{hn}n∈ℤ by G are, respectively, denoted by
(2.9){h̃n}n∈Z=G{hn}n∈Z,{hn′̃}n∈Z=G-{hn}n∈Z.
By directly calculating, we have the following results.Lemma 2.2.
If{hn}n∈ℤ is a one-dimensional CQF,  then {h̃n}n∈ℤ=G{hn}n∈ℤ and {hñ}n∈ℤ=G-{hn}n∈ℤ are one-dimensional CQFs.Lemma 2.3.
If{h̃n}n∈ℤ=G{hn}n∈ℤ,  then {hn}n∈ℤ=(GT)-{h̃n}n∈ℤ.Therefore, for a one-dimensional CQF, different one-dimensional CQFs can be derived when we choose different orthogonal matricesG. For a sequence {sn}n∈ℤ, letting  Γ={n:sn≠0}, we call Γ the support of {sn}n∈ℤ. If {sn}n∈ℤ is a one-dimensional CQF and the support of it is in {0,1}, we call {sn}n∈ℤ a simple one-dimensional CQF.Theorem 2.4.
If{hn}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{hn}n∈ℤ, we can get a simple one-dimensional CQF.
Without loss of generality, we assume that the support of{hn}n∈ℤ is in {0,1,2,…, 2N-1} and h0h2N-1≠0. We will prove this theorem by induction.
We see that the theorem is true forN=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 {hn}n∈ℤ is a one-dimensional CQF and h0h2L+1≠0. We denote that
(2.10)G=(cos⁡⁡θsin⁡θ-sin⁡θcos⁡⁡θ),{h̃n}n∈Z=G{hn}n∈Z,
where tanθ=-h1/h0.  Recall that {hn}n∈ℤ is a one-dimensional CQF, that is, ∑n∈ℤhnhn+2m=δ0,m, which implies that h0h2L+h1h2L+1=0. Therefore,
(2.11)h̃-1=sinθh0+cos⁡θh1=-cos⁡θh1+cos⁡θh1=0,h̃2L=cos⁡θh2L-sinθh2L+1=cos⁡θh2L+cos⁡θh1h0h2L+1=0.
It follows that the support of {h̃n}n∈ℤ is in {0,1,2,…,2L-1}. Suppose that -π/2<θ<π/2, then h̃0≠0. It follows that there exists t∈ℤ(t≤L), such that h̃0h̃2t-1≠0. Furthermore, if   n∈ℤ∖{0,1,2,…,2t-1},  then  h̃n=0.  By the hypothesis, the theorem is proved.Theorem2.4 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 {hn}n∈ℤ is a one-dimensional CQF. By Theorem 2.4, we know that {hn}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 Gθ as the 2×2 orthogonal matrix:
(2.12)(cos⁡θsinθ-sinθcos⁡θ),θ∈R.
Let {ℋn}n∈ℤγ be a simple one-dimensional CQF such that (2.13)Hn={cos⁡γ,n=0,sinγ,n=1,0,otherwise.γ∈R.
For an arbitrary positive integer N, choosing γ0,γ1,…,γN-1∈ℝ, we define {hnN}n∈ℤ as follows: (2.14){hnN}n∈Z=GγN-1GγN-2⋯Gγ1{Hn}n∈Zγ0.
In general, the support of {hnN}n∈ℤ is in [0,2N-1]∩ℤ. In other words, {hnN}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.
Letη=γ0+γ1+⋯+γN-1, then
(2.15)∑i∈Zh2iN=cos⁡⁡η,∑i∈Zh2i+1N=sin⁡η.Theorem 2.6.
Let(2.16)γ0+γ1+⋯+γN-1=2kπ+π4,k∈Z,
then the sequence {hnN}n∈ℤ given in  (2.14) is a one-dimensional low-pass wavelet filter.Theorem2.6 provides a condition for choosing γ0,γ1,…,γN-1 such that the one-dimensional CQF {hnN}n∈ℤ given in (2.14) is a one-dimensional orthogonal low-pass wavelet filter.In addition, if the low-pass wavelet filter{hnN}n∈ℤ satisfies the Cohen's condition (see [4]), then the φ(x) corresponding to {hnN}n∈ℤ in (2.2) is an orthogonal scaling function.By Theorem2.6, we know that a length-2N one-dimensional low-pass wavelet filter {hnN}n∈ℤ can be constructed by choosing γ0,γ1,…,γN-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: (2.17){hnN}n∈Z=GγN-1GγN-2⋯Gγ1{Hn}n∈Zπ/4-γN-1-γN-2-⋯-γ1,
where γ1,γ2,…,γN-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(3.1)ϕ(x)=2∑α∈Z2bαϕ(2x-α)
and its associated wavelets (3.2)ψl(x)=2∑α∈Z2dαlϕ(2x-α),l=1,2,3,
we need construct sequences {bα}α∈ℤ2 and {dαl}α∈ℤ2 such that (see [10, 12, 13], etc.) (3.3)∑α∈Z2bα=2,(3.4)∑α∈Z2bαbα+2β=δ0β,(3.5)∑i,j∈Zb(2i,2j)=∑i,j∈Zb(2i+1,2j)=∑i,j∈Zb(2i,2j+1)=∑i,j∈Zb(2i+1,2j+1),(3.6)∑α∈Z2dαj1dα+2βj2=δj1,j2δ0,β,(3.7)∑α∈Z2bαdα+2βj=0,
where j,j1,j2=1,2,3 and β∈ℤ2.The sequence{bα}α∈ℤ2, which satisfies (3.4), is called a two-dimensional CQF. If {bα}α∈ℤ2 satisfies (3.3), (3.4), and (3.5) simultaneously, we call it a two-dimensional low-pass wavelet filter. The sequences {dαj}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{lα}α∈ℤ2∈ℓ2(ℤ2), we define (3.8)Δ={α:α∈Z2,lα≠0},
and Δ is called the support set of {lα}α∈ℤ2∈ℓ2(ℤ2). If Δ is finite, we call {lα}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{bα}α∈ℤ2 can be split into 4 disjoint subsets
(3.9){b(2i,2j):i,j∈Z},{b(2i,2j+1):i,j∈Z},{b(2i+1,2j):i,j∈Z},{b(2i+1,2j+1):i,j∈Z}.Definition 3.1.
LetU be an arbitrary 4×4 orthogonal matrix, and let {bα}α∈ℤ2 be an FIR.For integersAs,Bs(s=1,2,3,4) and for all i,j∈ℤ, we set
(3.10)Γi,j̃=(b̃(2i,2j)b̃(2i,2j+1)b̃(2i+1,2j)b̃(2i+1,2j+1)))T,Γ(A1,B1,A2,B2,A3,B3,A4,B4)i,j=(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.
Then {b̃α}α∈ℤ2, which is defined as follows:
(3.11)Γi,j̃=Γ(A1,B1,A2,B2,A3,B3,A4,B4)i,jU
is called the two-dimensional SUT of {bα}α∈ℤ2.Lemma 3.2.
If{bα}α∈ℤ2 is a two-dimensional CQF and {b̃α}α∈ℤ2 is given by (3.11), then {b̃α}α∈ℤ2 is also a two-dimensional CQF.Proof.
By directly calculating, we can prove that{b̃α}α∈ℤ2 satisfies the following equation:
(3.12)∑α∈Z2b̃αb̃α+2β=δ0,β,∀β∈Z2.
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}α∈ℤ2 is a two-dimensional CQF,  {dαk}α∈ℤ2(k=1,2,3) satisfy  (3.6)  and  (3.7),  {b̃α}α∈ℤ2 is a two-dimensional CQF obtained by SUT of {bα}α∈ℤ2, then  {d̃αk}α∈ℤ2,  which are derived by SUT of  {dαk}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){b̃}α∈ℤ2 is called the  SUT0 of {bα}α∈ℤ2 if one chooses
(3.13)Ai=Bi=0(i=1,2,3,4)
in (3.11). This transform is denoted by {b̃α}α∈ℤ2=U0{bα}α∈ℤ2.
(ii){b̃}α∈ℤ2 is called the  SUTT1 of {bα}α∈ℤ2 if one chooses
(3.14)B2=B4=1,B1=B3=Ai=0(i=1,2,3,4)
in (3.11). This transform is denoted by {b̃α}α∈ℤ2=UT1{bα}α∈ℤ2.{b̃}α∈ℤ2 is called the  SUTT2 of {bα}α∈ℤ2 if one chooses
(3.15)A3=A4=1,A1=A2=Bi=0(i=1,2,3,4).
This transform is denoted by {b̃α}α∈ℤ2=UT2{bα}α∈ℤ2.
(iii){b̃}α∈ℤ2 is called the  SUT1 of {bα}α∈ℤ2 if one chooses
(3.16)A2=A4=1,A1=A3=Bi=0(i=1,2,3,4)
in (3.11). This transform is denoted by {b̃α}α∈ℤ2=U1{bα}α∈ℤ2.{b̃}α∈ℤ2 is called the  SUT2 of {bα}α∈ℤ2 if we choose
(3.17)B3=B4=1,B1=B2=Ai=0(i=1,2,3,4).
This transform is denoted by {b̃α}α∈ℤ2=U2{bα}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(3.18)bα={14ifα=(0,0)or(1,1),34ifα=(1,0)or(0,1),0otherwise
is {(0,0),(0,1),(1,0),(1,1)}, and the support of the filter {b̃α}α∈ℤ2=U1{bα}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
(3.19)H=[1232-3212].
In general, for integers N,M, if the support of {bα}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 {bα}α∈ℤ2 is {(0,0),(1,0),(0,1),(1,1)}, we call {bα}α∈ℤ2 a simple two-dimensional filter.We will adopt the following notations in the rest of this paper. For arbitraryξ0,λ0,ξ,λ∈R, let {ℬα}α∈ℤ2λ0,ξ0 be the FIR defined as follows:
(3.20)Bα={cos⁡ξ0cos⁡λ0,α=(0,0);cos⁡ξ0sinλ0,α=(0,1);sinξ0cos⁡λ0,α=(1,0);sinξ0sinλ0,α=(1,1);0,otherwise.
Furthermore, let {𝒟αk}α∈ℤ2λ0,ξ0(k=1,2,3) be the filters as follows
(3.21)Dα1={cos⁡ξ0sinλ0,α=(0,0);-cos⁡ξ0cos⁡λ0,α=(0,1);sinξ0sinλ0,α=(1,0);-sinξ0cos⁡λ0,α=(1,1);0,otherwise,Dα2={sinξ0cos⁡λ0,α=(0,0);sinξ0sinλ0,α=(0,1);-cos⁡ξ0cos⁡λ0,α=(1,0);-cos⁡ξ0sinλ0,α=(1,1);0,otherwise,(3.22)Dα3={sinξ0sinλ0,α=(0,0);-sinξ0cos⁡λ0,α=(0,1);-cos⁡ξ0sinλ0,α=(1,0);cos⁡ξ0cos⁡λ0,α=(1,1);0,otherwise,
and let Uλ,Uξ be orthogonal matrices such that (3.23)Uλ=(cos⁡λsinλ00-sinλcos⁡λ0000cos⁡λsinλ00-sinλcos⁡λ),Uξ=(cos⁡ξ0sinξ00cos⁡ξ0sinξ-sinξ0cos⁡ξ00-sinξ0cos⁡ξ),
respectively. Then {ℬα}α∈ℤ2λ0,ξ0 is a simple two-dimensional CQF, and {𝒟αk}α∈ℤ2λ0,ξ0 satisfy (3.6) and (3.7).From the special two-dimensional CQF{ℬα}α∈ℤ2λ0,ξ0, 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}α∈ℤ2 is called tensor-product wavelet filter if it satisfies the following equations: (3.24)b(2i,2j)=h2ih̃2j,b(2i,2j+1)=h2ih̃2j+1,b(2i+1,2j)=h2i+1h̃2j,b(2i+1,2j+1)=h2i+1h̃2j+1,i,j∈Z,
where {hi}i∈ℤ, {h̃i}i∈ℤ are one-dimensional orthogonal low-pass wavelet filters.Theorem 3.5.
If{bα}α∈ℤ2 is a tensor-product low-pass wavelet filter, then, it can be constructed by  SUTT1 and  SUTT2 from  {ℬα}α∈ℤ2λ0,ξ0.Proof.
Foralli,j∈ℤ, it follows from (3.24) that
(3.25)(b(2i,2j),b(2i,2j+1),b(2i+1,2j),b(2i+1,2j+1))=(h2i,h2i,h2i+1,h2i+1)(h̃2j0000h̃2j+10000h̃2j0000h̃2j+1).
Suppose that {h̃i}i∈ℤ is a length-2N one-dimensional filter. Then there exist λ0,…,λN-2,λN-1∈R such that λ0+⋯+λN-2+λN-1=π/4, and
(3.26){h̃n}n∈Z=GλN-1GλN-2⋯Gλ1{Hn}n∈Zλ0.
We denote
(3.27){h̃n}n∈ZN-1=GλN-2⋯Gλ1{Hn}n∈Zλ0.
Let {bα}α∈ℤ2N-1 be two-dimensional filter such that
(3.28)b(2i,2j)N-1=h2ih̃2jN-1,b(2i,2j+1)N-1=h2ih̃2j+1N-1,b(2i+1,2j)N-1=h2i+1h̃2jN-1,b(2i+1,2j+1)N-1=h2i+1h̃2j+1N-1,i,j∈Z.
It follows that
(3.29)(b(2i,2j),b(2i,2j+1),b(2i+1,2j),b(2i+1,2j+1))=(h2i,h2i,h2i+1,h2i+1)(h̃2j0000h̃2j+10000h̃2j0000h̃2j+1)=(h2i,h2i,h2i+1,h2i+1)(h̃2jN-10000h̃2j-1N-10000h̃2jN-10000h̃2j-1N-1)×(cos⁡λN-1sinλN-100-sinλN-1cos⁡λN-10000cos⁡λN-1sinλN-100-sinλN-1cos⁡λN-1)=(b(2i,2j)N-1,b(2i,2j-1)N-1,b(2i+1,2j)N-1,b(2i+1,2j-1)N-1)UλN-1.
Hence, {bα}α∈ℤ2 can be constructed by SUTT1 of {bα}α∈ℤ2N-1.
It can be proved inductively that{bα}α∈ℤ2 can be constructed by a series of SUTT1 from {bα}α∈ℤ20, and {bα}α∈ℤ20 can be constructed by a series of SUTT2 from {ℬα}α∈ℤ2λ0,ξ0, where {bα}α∈ℤ20 denotes the filter
(3.30)b(2i,2j)0=h2iH̃2j,b(2i,2j+1)0=h2iH̃2j+1,b(2i+1,2j)0=h2i+1H̃2j,b(2i+1,2j+1)0=h2i+1H̃2j+1,i,j∈Z.
Therefore, any tensor-product two-dimensional low-pass wavelet filters can be constructed by SUTT1 and SUTT2 from {ℬα}α∈ℤ2λ0,ξ0.


### 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 integersN and M, choosing λ0,λ1,…,λN-1,ξ0,ξ1,…,ξM-1∈ℝ, {bαN,M}α∈ℤ2 is defined as (3.31){bαN,M}α∈Z2=UεnN+M-2nN+M-2UεnN+M-3nN+M-3⋯Un1εn1{Bα}α∈Z2λ0,ξ0,
where n1,n2,…,nN+M-2∈{λ1,λ2,…,λN-1,ξ1,ξ2,…,ξM-1} and (3.32)εnj={1,ifnj∈{λ1,λ2,…,λN-1},2,ifnj∈{ξ1,ξ2,…,ξM-1}.
Then {bαN,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,…,λN1,ξ0,ξ1,…,ξM1, such that  {bαN,M}α∈ℤ2 is a two-dimensional low-pass wavelet filter.Theorem 3.6.
Letη=λ0+λ1+⋯+λN-1,η̃=ξ0+ξ1+⋯+ξM-1.Then
(3.33)∑i,j∈Zb(2i,2j)N,M=cos⁡ηcos⁡η̃,∑i,j∈Zb(2i,2j+1)N,M=sinηcos⁡η̃,(3.34)∑i,j∈Zb(2i+1,2j)N,M=cos⁡ηsinη̃,∑i,j∈Zb(2i+1,2j+1)N,M=sinηsinη̃.Proof.
It can be proved inductively. For the caseN=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.
Lets be the integer such that ns+1∈{λ1,λ2,…,λk1} and ns+t∈{ξ1,ξ2,…,ξk2}(t>1,t∈ℤ). Suppose that
(3.35)i1,i2,…,iu≤s⟹ni1,ni2,…,niu∈{λ1,λ2,…,λk1},j1,j2,…,jv≤s⟹nj1,nj2,…,njv∈{ξ1,ξ2,…,ξk1}.
It follows that u=k1 and v≤k2-1. Let
(3.36)ηs=λ0+ni1+ni2+⋯+niu,η̃s=ξ0+nj1+nj2+⋯+njv,{bαs}α∈Z2=UεnsnsUεns-1ns-1⋯Uεn1n1{Bα}α∈Z2λ0,ξ0;
it follows that
(3.37)∑i,j∈Zb(2i,2j)s=cos⁡ηscos⁡η̃s,∑i,j∈Zb(2i,2j+1)s=sinηscos⁡η̃s,∑i,j∈Zb(2i+1,2j)s=cos⁡ηssinη̃s,∑i,j∈Zb(2i+1,2j+1)s=sinηssinη̃s.
Let {bαs+1}α∈ℤ2=Uεns+1ns+1{bαs}α∈ℤ2. Since ns+1∈{λ1,λ2,…,λk1}, then
(3.38)b(2i,2j)s+1=cos⁡ns+1b(2i,2j)s-sinns+1b(2i-2,2j+1)s,b(2i,2j+1)s+1=sinns+1b(2i,2j)s+cos⁡ns+1b(2i-2,2j+1)s,b(2i+1,2j)s+1=cos⁡ns+1b(2i+1,2j)s-sinns+1b(2i-1,2j+1)s,b(2i+1,2j+1)s+1=sinns+1b(2i+1,2j)s+cos⁡ns+1b(2i-1,2j+1)s.
Therefore,
(3.39)∑i,j∈Zb(2i,2j)s+1=cos⁡(ηs+ns+1)cos⁡η̃s;∑i,j∈Zb(2i,2j+1)s+1=sin(ηs+ns+1)cos⁡η̃s;∑i,j∈Zb(2i+1,2j)s+1=cos⁡(ηs+ns+1)sinη̃s;∑i,j∈Zb(2i+1,2j+1)s+1=sin(ηs+ns+1)sinη̃s.
For t>1, let
(3.40){bαs+t}α∈Z2=Uεns+tns+t⋯Uεns+1ns+1{bαs}α∈Z2.
It follows from ns+t∈{ξ1,ξ2,…,ξk2-1} that
(3.41)b(2i,2j)s+t=cos⁡ns+tb(2i,2j)s+t-1-sinns+tb(2i+1,2j-2)s+t-1,b(2i,2j+1)s+t=cos⁡ns+tb(2i,2j+1)s+t-1-sinns+tb(2i+1,2j-1)s+t-1,b(2i+1,2j)s+t=sinns+tb(2i,2j)s+t-1+cos⁡ns+tb(2i+1,2j-2)s+t-1,b(2i+1,2j+1)s+t=sinns+tb(2i,2j+1)s+t-1+cos⁡ns+tb(2i+1,2j-1)s+t-1.
Therefore,
(3.42)∑i,j∈Zb(2i,2j)s+t=cos⁡ηs+t-1cos⁡(η̃s+t-1+ns+t);∑i,j∈Zb(2i,2j+1)s+t=sinηs+t-1cos⁡(η̃s+t-1+ns+t);∑i,j∈Zb(2i+1,2j)s+t=cos⁡ηs+t-1sin(η̃s+t-1+ns+t);∑i,j∈Zb(2i+1,2j+1)s+t=sinηs+t-1sin(η̃s+t-1+ns+t),
where
(3.43)ηs+t-1=λ0+λ1+⋯+λk1,η̃s+t-1=ξ0+nj1+⋯+njv+ns+2+ns+t-1.
Namely, it is true for the case N=k1+1,M=k2.
Similarly for the caseN=k1,M=k2+1. This completes the proof.Now we provide the condition on choosingλ0,λ1,…,λN1,ξ0,ξ1,…,ξM1, such that {bαN,M}α∈ℤ2 is a two-dimensional low-pass wavelet filter.Theorem 3.7.
If there exists integersn1 and  n2, such that
(3.44)η=λ0+λ1+⋯+λN-1=2n1π+π4,(3.45)η̃=ξ0+ξ1+⋯+ξM-1=2n2π+π4,
then the sequence {bαN,M}α∈ℤ2 constructed in  (3.31) is a two-dimensional low-pass wavelet filter.Proof.
It follows from Theorem3.6 that
(3.46)∑i,j∈Zb(2i,2j)=cos⁡ηcos⁡η̃=12,∑i,j∈Zb(2i,2j+1)=sinηcos⁡η̃=12,∑i,j∈Zb(2i+1,2j)=cos⁡ηsinη̃=12,∑i,j∈Zb(2i+1,2j+1)=sinηsinη̃=12.
Therefore, {bαN,M}α∈ℤ2 satisfies conditions (3.3) and (3.5), then it is a two-dimensional low-pass wavelet filter.Corollary 3.8.
If{bαN,M}α∈ℤ2 as given in  (3.31) is a low-pass wavelet filter, then
(3.47){dαk,N,M}α∈Z2=UεnN+M-2nN+M-2UεnN+M-3nN+M-3⋯Uεn1n1{Dαk}α∈Z2λ0,ξ0
are the high-pass filters associated with {bαN,M}α∈ℤ2, where k=1,2,3.Remark 3.9.
We can choose other orthogonal matrices thanUλ and Uξ such as
(3.48)Uλ,λ̃=(cos⁡λsinλ00-sinλcos⁡λ0000cos⁡λ̃sinλ̃00-sinλ̃cos⁡λ̃),Uξ,ξ̃=(cos⁡ξ0sinξ00cos⁡ξ̃0sinξ̃-sinξ0cos⁡ξ00-sinξ̃0cos⁡ξ̃),
where λ,λ̃,ξ,ξ̃∈ℝ; but conditions in Theorem 3.7 should be changed.Example 3.10.
We chooseN=2,M=2in (3.31). Forξ0,ξ1,λ0,λ1∈ℝ, define{bα}α∈ℤ2as follows:(3.49){bα}α∈Z2=Uξ12Uλ11{Bα}α∈Z2λ0,ξ0.By choosingξ0,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.
b(i,j)j=0j=1j=2j=3i=0−0.020275−0.0547870.2432500.657306i=10.0248990.0672820.1980760.535237i=2−0.0251900.0093220.302214−0.111841i=30.030935−0.0114480.246090−0.091071Table 2
The coefficients of a high-pass wavelet filter (d(i,j)1) associated with the low-pass wavelet filter in Table 1.
d(i,j)1j=0j=1j=2j=3i=00.0093220.025190−0.111841−0.302214i=1−0.011448−0.030935−0.091071−0.246090i=2−0.0547870.0202750.657306−0.243250i=30.067282−0.0248990.535237−0.198076Table 3
The coefficients of a high-pass wavelet filter (d(i,j)2) associated with the low-pass wavelet filter in Table 1.
d(i,j)2j=0j=1j=2j=3i=00.1980760.5352370.0248990.067282i=1−0.243250−0.6573060.0202750.054787i=20.246090−0.0910710.030935−0.011448i=3−0.3022140.1118410.025190−0.009322Table 4
The coefficients of a high-pass wavelet filter (d(i,j)3) associated with the low-pass wavelet filter in Table 1.
d(i,j)3j=0j=1j=2j=3i=0−0.091071−0.246090−0.011448−0.030935i=10.1118410.302214−0.009322−0.025190i=20.535237−0.1980760.067282−0.024899i=3−0.6573060.2432500.054787−0.020275It 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.
(a)(b)Remark 3.11.
By Section3.2, 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.


## 3.1. SUT of Two-Dimensional CQF

To construct two-dimensional orthogonal scaling function(3.1)ϕ(x)=2∑α∈Z2bαϕ(2x-α)
and its associated wavelets (3.2)ψl(x)=2∑α∈Z2dαlϕ(2x-α),l=1,2,3,
we need construct sequences {bα}α∈ℤ2 and {dαl}α∈ℤ2 such that (see [10, 12, 13], etc.) (3.3)∑α∈Z2bα=2,(3.4)∑α∈Z2bαbα+2β=δ0β,(3.5)∑i,j∈Zb(2i,2j)=∑i,j∈Zb(2i+1,2j)=∑i,j∈Zb(2i,2j+1)=∑i,j∈Zb(2i+1,2j+1),(3.6)∑α∈Z2dαj1dα+2βj2=δj1,j2δ0,β,(3.7)∑α∈Z2bαdα+2βj=0,
where j,j1,j2=1,2,3 and β∈ℤ2.The sequence{bα}α∈ℤ2, which satisfies (3.4), is called a two-dimensional CQF. If {bα}α∈ℤ2 satisfies (3.3), (3.4), and (3.5) simultaneously, we call it a two-dimensional low-pass wavelet filter. The sequences {dαj}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{lα}α∈ℤ2∈ℓ2(ℤ2), we define (3.8)Δ={α:α∈Z2,lα≠0},
and Δ is called the support set of {lα}α∈ℤ2∈ℓ2(ℤ2). If Δ is finite, we call {lα}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{bα}α∈ℤ2 can be split into 4 disjoint subsets
(3.9){b(2i,2j):i,j∈Z},{b(2i,2j+1):i,j∈Z},{b(2i+1,2j):i,j∈Z},{b(2i+1,2j+1):i,j∈Z}.Definition 3.1.
LetU be an arbitrary 4×4 orthogonal matrix, and let {bα}α∈ℤ2 be an FIR.For integersAs,Bs(s=1,2,3,4) and for all i,j∈ℤ, we set
(3.10)Γi,j̃=(b̃(2i,2j)b̃(2i,2j+1)b̃(2i+1,2j)b̃(2i+1,2j+1)))T,Γ(A1,B1,A2,B2,A3,B3,A4,B4)i,j=(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.
Then {b̃α}α∈ℤ2, which is defined as follows:
(3.11)Γi,j̃=Γ(A1,B1,A2,B2,A3,B3,A4,B4)i,jU
is called the two-dimensional SUT of {bα}α∈ℤ2.Lemma 3.2.
If{bα}α∈ℤ2 is a two-dimensional CQF and {b̃α}α∈ℤ2 is given by (3.11), then {b̃α}α∈ℤ2 is also a two-dimensional CQF.Proof.
By directly calculating, we can prove that{b̃α}α∈ℤ2 satisfies the following equation:
(3.12)∑α∈Z2b̃αb̃α+2β=δ0,β,∀β∈Z2.
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}α∈ℤ2 is a two-dimensional CQF,  {dαk}α∈ℤ2(k=1,2,3) satisfy  (3.6)  and  (3.7),  {b̃α}α∈ℤ2 is a two-dimensional CQF obtained by SUT of {bα}α∈ℤ2, then  {d̃αk}α∈ℤ2,  which are derived by SUT of  {dαk}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){b̃}α∈ℤ2 is called the  SUT0 of {bα}α∈ℤ2 if one chooses
(3.13)Ai=Bi=0(i=1,2,3,4)
in (3.11). This transform is denoted by {b̃α}α∈ℤ2=U0{bα}α∈ℤ2.
(ii){b̃}α∈ℤ2 is called the  SUTT1 of {bα}α∈ℤ2 if one chooses
(3.14)B2=B4=1,B1=B3=Ai=0(i=1,2,3,4)
in (3.11). This transform is denoted by {b̃α}α∈ℤ2=UT1{bα}α∈ℤ2.{b̃}α∈ℤ2 is called the  SUTT2 of {bα}α∈ℤ2 if one chooses
(3.15)A3=A4=1,A1=A2=Bi=0(i=1,2,3,4).
This transform is denoted by {b̃α}α∈ℤ2=UT2{bα}α∈ℤ2.
(iii){b̃}α∈ℤ2 is called the  SUT1 of {bα}α∈ℤ2 if one chooses
(3.16)A2=A4=1,A1=A3=Bi=0(i=1,2,3,4)
in (3.11). This transform is denoted by {b̃α}α∈ℤ2=U1{bα}α∈ℤ2.{b̃}α∈ℤ2 is called the  SUT2 of {bα}α∈ℤ2 if we choose
(3.17)B3=B4=1,B1=B2=Ai=0(i=1,2,3,4).
This transform is denoted by {b̃α}α∈ℤ2=U2{bα}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(3.18)bα={14ifα=(0,0)or(1,1),34ifα=(1,0)or(0,1),0otherwise
is {(0,0),(0,1),(1,0),(1,1)}, and the support of the filter {b̃α}α∈ℤ2=U1{bα}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
(3.19)H=[1232-3212].
In general, for integers N,M, if the support of {bα}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 {bα}α∈ℤ2 is {(0,0),(1,0),(0,1),(1,1)}, we call {bα}α∈ℤ2 a simple two-dimensional filter.We will adopt the following notations in the rest of this paper. For arbitraryξ0,λ0,ξ,λ∈R, let {ℬα}α∈ℤ2λ0,ξ0 be the FIR defined as follows:
(3.20)Bα={cos⁡ξ0cos⁡λ0,α=(0,0);cos⁡ξ0sinλ0,α=(0,1);sinξ0cos⁡λ0,α=(1,0);sinξ0sinλ0,α=(1,1);0,otherwise.
Furthermore, let {𝒟αk}α∈ℤ2λ0,ξ0(k=1,2,3) be the filters as follows
(3.21)Dα1={cos⁡ξ0sinλ0,α=(0,0);-cos⁡ξ0cos⁡λ0,α=(0,1);sinξ0sinλ0,α=(1,0);-sinξ0cos⁡λ0,α=(1,1);0,otherwise,Dα2={sinξ0cos⁡λ0,α=(0,0);sinξ0sinλ0,α=(0,1);-cos⁡ξ0cos⁡λ0,α=(1,0);-cos⁡ξ0sinλ0,α=(1,1);0,otherwise,(3.22)Dα3={sinξ0sinλ0,α=(0,0);-sinξ0cos⁡λ0,α=(0,1);-cos⁡ξ0sinλ0,α=(1,0);cos⁡ξ0cos⁡λ0,α=(1,1);0,otherwise,
and let Uλ,Uξ be orthogonal matrices such that (3.23)Uλ=(cos⁡λsinλ00-sinλcos⁡λ0000cos⁡λsinλ00-sinλcos⁡λ),Uξ=(cos⁡ξ0sinξ00cos⁡ξ0sinξ-sinξ0cos⁡ξ00-sinξ0cos⁡ξ),
respectively. Then {ℬα}α∈ℤ2λ0,ξ0 is a simple two-dimensional CQF, and {𝒟αk}α∈ℤ2λ0,ξ0 satisfy (3.6) and (3.7).From the special two-dimensional CQF{ℬα}α∈ℤ2λ0,ξ0, 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}α∈ℤ2 is called tensor-product wavelet filter if it satisfies the following equations: (3.24)b(2i,2j)=h2ih̃2j,b(2i,2j+1)=h2ih̃2j+1,b(2i+1,2j)=h2i+1h̃2j,b(2i+1,2j+1)=h2i+1h̃2j+1,i,j∈Z,
where {hi}i∈ℤ, {h̃i}i∈ℤ are one-dimensional orthogonal low-pass wavelet filters.Theorem 3.5.
If{bα}α∈ℤ2 is a tensor-product low-pass wavelet filter, then, it can be constructed by  SUTT1 and  SUTT2 from  {ℬα}α∈ℤ2λ0,ξ0.Proof.
Foralli,j∈ℤ, it follows from (3.24) that
(3.25)(b(2i,2j),b(2i,2j+1),b(2i+1,2j),b(2i+1,2j+1))=(h2i,h2i,h2i+1,h2i+1)(h̃2j0000h̃2j+10000h̃2j0000h̃2j+1).
Suppose that {h̃i}i∈ℤ is a length-2N one-dimensional filter. Then there exist λ0,…,λN-2,λN-1∈R such that λ0+⋯+λN-2+λN-1=π/4, and
(3.26){h̃n}n∈Z=GλN-1GλN-2⋯Gλ1{Hn}n∈Zλ0.
We denote
(3.27){h̃n}n∈ZN-1=GλN-2⋯Gλ1{Hn}n∈Zλ0.
Let {bα}α∈ℤ2N-1 be two-dimensional filter such that
(3.28)b(2i,2j)N-1=h2ih̃2jN-1,b(2i,2j+1)N-1=h2ih̃2j+1N-1,b(2i+1,2j)N-1=h2i+1h̃2jN-1,b(2i+1,2j+1)N-1=h2i+1h̃2j+1N-1,i,j∈Z.
It follows that
(3.29)(b(2i,2j),b(2i,2j+1),b(2i+1,2j),b(2i+1,2j+1))=(h2i,h2i,h2i+1,h2i+1)(h̃2j0000h̃2j+10000h̃2j0000h̃2j+1)=(h2i,h2i,h2i+1,h2i+1)(h̃2jN-10000h̃2j-1N-10000h̃2jN-10000h̃2j-1N-1)×(cos⁡λN-1sinλN-100-sinλN-1cos⁡λN-10000cos⁡λN-1sinλN-100-sinλN-1cos⁡λN-1)=(b(2i,2j)N-1,b(2i,2j-1)N-1,b(2i+1,2j)N-1,b(2i+1,2j-1)N-1)UλN-1.
Hence, {bα}α∈ℤ2 can be constructed by SUTT1 of {bα}α∈ℤ2N-1.
It can be proved inductively that{bα}α∈ℤ2 can be constructed by a series of SUTT1 from {bα}α∈ℤ20, and {bα}α∈ℤ20 can be constructed by a series of SUTT2 from {ℬα}α∈ℤ2λ0,ξ0, where {bα}α∈ℤ20 denotes the filter
(3.30)b(2i,2j)0=h2iH̃2j,b(2i,2j+1)0=h2iH̃2j+1,b(2i+1,2j)0=h2i+1H̃2j,b(2i+1,2j+1)0=h2i+1H̃2j+1,i,j∈Z.
Therefore, any tensor-product two-dimensional low-pass wavelet filters can be constructed by SUTT1 and SUTT2 from {ℬα}α∈ℤ2λ0,ξ0.


## 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 integersN and M, choosing λ0,λ1,…,λN-1,ξ0,ξ1,…,ξM-1∈ℝ, {bαN,M}α∈ℤ2 is defined as (3.31){bαN,M}α∈Z2=UεnN+M-2nN+M-2UεnN+M-3nN+M-3⋯Un1εn1{Bα}α∈Z2λ0,ξ0,
where n1,n2,…,nN+M-2∈{λ1,λ2,…,λN-1,ξ1,ξ2,…,ξM-1} and (3.32)εnj={1,ifnj∈{λ1,λ2,…,λN-1},2,ifnj∈{ξ1,ξ2,…,ξM-1}.
Then {bαN,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,…,λN1,ξ0,ξ1,…,ξM1, such that  {bαN,M}α∈ℤ2 is a two-dimensional low-pass wavelet filter.Theorem 3.6.
Letη=λ0+λ1+⋯+λN-1,η̃=ξ0+ξ1+⋯+ξM-1.Then
(3.33)∑i,j∈Zb(2i,2j)N,M=cos⁡ηcos⁡η̃,∑i,j∈Zb(2i,2j+1)N,M=sinηcos⁡η̃,(3.34)∑i,j∈Zb(2i+1,2j)N,M=cos⁡ηsinη̃,∑i,j∈Zb(2i+1,2j+1)N,M=sinηsinη̃.Proof.
It can be proved inductively. For the caseN=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.
Lets be the integer such that ns+1∈{λ1,λ2,…,λk1} and ns+t∈{ξ1,ξ2,…,ξk2}(t>1,t∈ℤ). Suppose that
(3.35)i1,i2,…,iu≤s⟹ni1,ni2,…,niu∈{λ1,λ2,…,λk1},j1,j2,…,jv≤s⟹nj1,nj2,…,njv∈{ξ1,ξ2,…,ξk1}.
It follows that u=k1 and v≤k2-1. Let
(3.36)ηs=λ0+ni1+ni2+⋯+niu,η̃s=ξ0+nj1+nj2+⋯+njv,{bαs}α∈Z2=UεnsnsUεns-1ns-1⋯Uεn1n1{Bα}α∈Z2λ0,ξ0;
it follows that
(3.37)∑i,j∈Zb(2i,2j)s=cos⁡ηscos⁡η̃s,∑i,j∈Zb(2i,2j+1)s=sinηscos⁡η̃s,∑i,j∈Zb(2i+1,2j)s=cos⁡ηssinη̃s,∑i,j∈Zb(2i+1,2j+1)s=sinηssinη̃s.
Let {bαs+1}α∈ℤ2=Uεns+1ns+1{bαs}α∈ℤ2. Since ns+1∈{λ1,λ2,…,λk1}, then
(3.38)b(2i,2j)s+1=cos⁡ns+1b(2i,2j)s-sinns+1b(2i-2,2j+1)s,b(2i,2j+1)s+1=sinns+1b(2i,2j)s+cos⁡ns+1b(2i-2,2j+1)s,b(2i+1,2j)s+1=cos⁡ns+1b(2i+1,2j)s-sinns+1b(2i-1,2j+1)s,b(2i+1,2j+1)s+1=sinns+1b(2i+1,2j)s+cos⁡ns+1b(2i-1,2j+1)s.
Therefore,
(3.39)∑i,j∈Zb(2i,2j)s+1=cos⁡(ηs+ns+1)cos⁡η̃s;∑i,j∈Zb(2i,2j+1)s+1=sin(ηs+ns+1)cos⁡η̃s;∑i,j∈Zb(2i+1,2j)s+1=cos⁡(ηs+ns+1)sinη̃s;∑i,j∈Zb(2i+1,2j+1)s+1=sin(ηs+ns+1)sinη̃s.
For t>1, let
(3.40){bαs+t}α∈Z2=Uεns+tns+t⋯Uεns+1ns+1{bαs}α∈Z2.
It follows from ns+t∈{ξ1,ξ2,…,ξk2-1} that
(3.41)b(2i,2j)s+t=cos⁡ns+tb(2i,2j)s+t-1-sinns+tb(2i+1,2j-2)s+t-1,b(2i,2j+1)s+t=cos⁡ns+tb(2i,2j+1)s+t-1-sinns+tb(2i+1,2j-1)s+t-1,b(2i+1,2j)s+t=sinns+tb(2i,2j)s+t-1+cos⁡ns+tb(2i+1,2j-2)s+t-1,b(2i+1,2j+1)s+t=sinns+tb(2i,2j+1)s+t-1+cos⁡ns+tb(2i+1,2j-1)s+t-1.
Therefore,
(3.42)∑i,j∈Zb(2i,2j)s+t=cos⁡ηs+t-1cos⁡(η̃s+t-1+ns+t);∑i,j∈Zb(2i,2j+1)s+t=sinηs+t-1cos⁡(η̃s+t-1+ns+t);∑i,j∈Zb(2i+1,2j)s+t=cos⁡ηs+t-1sin(η̃s+t-1+ns+t);∑i,j∈Zb(2i+1,2j+1)s+t=sinηs+t-1sin(η̃s+t-1+ns+t),
where
(3.43)ηs+t-1=λ0+λ1+⋯+λk1,η̃s+t-1=ξ0+nj1+⋯+njv+ns+2+ns+t-1.
Namely, it is true for the case N=k1+1,M=k2.
Similarly for the caseN=k1,M=k2+1. This completes the proof.Now we provide the condition on choosingλ0,λ1,…,λN1,ξ0,ξ1,…,ξM1, such that {bαN,M}α∈ℤ2 is a two-dimensional low-pass wavelet filter.Theorem 3.7.
If there exists integersn1 and  n2, such that
(3.44)η=λ0+λ1+⋯+λN-1=2n1π+π4,(3.45)η̃=ξ0+ξ1+⋯+ξM-1=2n2π+π4,
then the sequence {bαN,M}α∈ℤ2 constructed in  (3.31) is a two-dimensional low-pass wavelet filter.Proof.
It follows from Theorem3.6 that
(3.46)∑i,j∈Zb(2i,2j)=cos⁡ηcos⁡η̃=12,∑i,j∈Zb(2i,2j+1)=sinηcos⁡η̃=12,∑i,j∈Zb(2i+1,2j)=cos⁡ηsinη̃=12,∑i,j∈Zb(2i+1,2j+1)=sinηsinη̃=12.
Therefore, {bαN,M}α∈ℤ2 satisfies conditions (3.3) and (3.5), then it is a two-dimensional low-pass wavelet filter.Corollary 3.8.
If{bαN,M}α∈ℤ2 as given in  (3.31) is a low-pass wavelet filter, then
(3.47){dαk,N,M}α∈Z2=UεnN+M-2nN+M-2UεnN+M-3nN+M-3⋯Uεn1n1{Dαk}α∈Z2λ0,ξ0
are the high-pass filters associated with {bαN,M}α∈ℤ2, where k=1,2,3.Remark 3.9.
We can choose other orthogonal matrices thanUλ and Uξ such as
(3.48)Uλ,λ̃=(cos⁡λsinλ00-sinλcos⁡λ0000cos⁡λ̃sinλ̃00-sinλ̃cos⁡λ̃),Uξ,ξ̃=(cos⁡ξ0sinξ00cos⁡ξ̃0sinξ̃-sinξ0cos⁡ξ00-sinξ̃0cos⁡ξ̃),
where λ,λ̃,ξ,ξ̃∈ℝ; but conditions in Theorem 3.7 should be changed.Example 3.10.
We chooseN=2,M=2in (3.31). Forξ0,ξ1,λ0,λ1∈ℝ, define{bα}α∈ℤ2as follows:(3.49){bα}α∈Z2=Uξ12Uλ11{Bα}α∈Z2λ0,ξ0.By choosingξ0,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.
b(i,j)j=0j=1j=2j=3i=0−0.020275−0.0547870.2432500.657306i=10.0248990.0672820.1980760.535237i=2−0.0251900.0093220.302214−0.111841i=30.030935−0.0114480.246090−0.091071Table 2
The coefficients of a high-pass wavelet filter (d(i,j)1) associated with the low-pass wavelet filter in Table 1.
d(i,j)1j=0j=1j=2j=3i=00.0093220.025190−0.111841−0.302214i=1−0.011448−0.030935−0.091071−0.246090i=2−0.0547870.0202750.657306−0.243250i=30.067282−0.0248990.535237−0.198076Table 3
The coefficients of a high-pass wavelet filter (d(i,j)2) associated with the low-pass wavelet filter in Table 1.
d(i,j)2j=0j=1j=2j=3i=00.1980760.5352370.0248990.067282i=1−0.243250−0.6573060.0202750.054787i=20.246090−0.0910710.030935−0.011448i=3−0.3022140.1118410.025190−0.009322Table 4
The coefficients of a high-pass wavelet filter (d(i,j)3) associated with the low-pass wavelet filter in Table 1.
d(i,j)3j=0j=1j=2j=3i=0−0.091071−0.246090−0.011448−0.030935i=10.1118410.302214−0.009322−0.025190i=20.535237−0.1980760.067282−0.024899i=3−0.6573060.2432500.054787−0.020275It 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.
(a)(b)Remark 3.11.
By Section3.2, 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.



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