Shift Unitary Transform for Constructing Two-Dimensional Wavelet Filters

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.


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 tensorproduct 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

Parametrization of One-Dimensional Wavelet Filters
where for any i ∈ Z,

2.8
The SUT and the ISUT of {h n } n∈Z by G are, respectively, denoted by By directly calculating, we have the following results.
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.
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 onedimensional CQF and h 0 h 2L 1 / 0. We denote that where

2.11
It follows that the support of 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:

2.12
Let {H n } γ n∈Z be a simple one-dimensional CQF such that γ ∈ R.

2.14
In general, the support of {h N n } n∈Z is in 0, 2N − 1 ∩ Z.In other words, {h N n } 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.

2.15
Theorem 2.6.Let then the sequence {h N n } n∈Z given in 2.14 is a one-dimensional low-pass wavelet filter.
Theorem 2.6 provides a condition for choosing γ 0 , γ 1 , . . ., γ N−1 such that the one-dimensional CQF {h N n } n∈Z given in 2.14 is a one-dimensional orthogonal low-pass wavelet filter.In addition, if the low-pass wavelet filter {h N n } n∈Z satisfies the Cohen's condition see 4 , then the ϕ x corresponding to {h N n } 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 N n } n∈Z can be constructed by choosing γ 0 , γ 1 , . . ., 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: where 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.

SUT of Two-Dimensional CQF
To construct two-dimensional orthogonal scaling function and its associated wavelets we need construct sequences {b α } α∈Z 2 and {d l α } α∈Z 2 such that see 10, 12, 13 , etc.
For an arbitrary real-valued sequence {l α } α∈Z 2 ∈ 2 Z 2 , we define As aforementioned, the sequences we consider are real-valued and finite supported FIR .We note that any sequence {b α } α∈Z 2 can be split into 4 disjoint subsets

3.9
Definition 3.1.Let U be an arbitrary 4 × 4 orthogonal matrix, and let {b α } α∈Z 2 be an FIR.For integers A s , B s s 1, 2, 3, 4 and for all i, j ∈ Z, we set

3.10
Then { b α } α∈Z 2 , which is defined as follows: Proof.By directly calculating, we can prove that { b α } α∈Z 2 satisfies the following equation:

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.
This transform is denoted by 3.17 This transform is denoted by 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 In general, for integers N, M, if the support of We will adopt the following notations in the rest of this paper.For arbitrary ξ 0 , λ 0 , ξ, λ ∈ R, let {B α } λ 0 ,ξ 0 α∈Z 2 be the FIR defined as follows: sin ξ 0 cos λ 0 , α 1, 0 ; sin ξ 0 sin λ 0 , α 1, 1 ; 0, otherwise.

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∈Z 2 is called tensor-product wavelet filter if it satisfies the following equations: Proof.For all i, j ∈ Z, it follows from 3.24 that

3.26
We denote

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

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 Let s be the integer such that n s 1 ∈ {λ 1 , λ 2 , . . ., λ k 1 } and

3.35
It follows that u k 1 and v ≤ k 2 − 1.Let sin η s sin η s .

3.43
Namely, it is true for the case N k 1 1, M k 2 .
Similarly for the case N k 1 , M k 2 1.This completes the proof.

3.49
By choosing ξ 0 , ξ 1 , λ 0 , λ 1 such that λ 0 λ 1 ξ 0 ξ 1 π/4, we can get many twodimensional low-pass wavelet filters and their corresponding high-pass wavelet filters.For instance, set ξ 1 2.254190, ξ 0 π/4 − ξ 1 , λ 1 4.357946, λ 0 π/4 − λ 1 .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 subbands by the derived filter can reveal more features than that by the commonly used tensorproduct wavelet filter.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.

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.

Figure 1 :
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.

Table 1 :
The coefficients of a nonseparable orthogonal low-pass wavelet filter.

Table 2 :
The coefficients of a high-pass wavelet filter d1

Table 3 :
The coefficients of a high-pass wavelet filter d 2 i,j associated with the low-pass wavelet filter in

Table 4 :
The coefficients of a high-pass wavelet filter d3