Research Article  Open Access
Fei Li, Jianwei Yang, "Shift Unitary Transform for Constructing TwoDimensional Wavelet Filters", Journal of Applied Mathematics, vol. 2011, Article ID 272801, 19 pages, 2011. https://doi.org/10.1155/2011/272801
Shift Unitary Transform for Constructing TwoDimensional Wavelet Filters
Abstract
Due to the difficulty for constructing twodimensional wavelet filters, the commonly used wavelet filters are tensorproduct of onedimensional 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 twodimensional orthogonal wavelet filters. It is proved that tensorproduct wavelet filters are only special cases of this parametrization method. To show this, we introduce the SUT of onedimensional CQF and present a complete parametrization of onedimensional wavelet system. As a result, more ways are provided to randomly generate twodimensional 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 onedimensional 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, twodimensional wavelets have to be constructed. The most common wavelets used for image processing are tensorproduct of onedimensional 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 Kovaević [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 waveletbased watermarking [17, 18], parametrization of twodimensional wavelet filters is preferred. To make some waveletbased 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 twodimensional wavelet filters, one has to solve transcendental constraints for the parameters. Hence, wavelet filters used in waveletbased watermarking schemes, such as [17–20], are only tensorproduct of onedimensional 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 twodimensional orthogonal wavelet filters. The choosing of the parameters is not restricted by any implicit condition. It is proved that tensorproduct 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 , we denote its transpose by in this paper.
To show that tensorproduct wavelet filters are only special cases of our construction, we introduce the SUT of onedimensional CQF and present a complete parametrization of onedimensional wavelet system in Section 2. In Section 3, we show that tensorproduct orthogonal wavelet filters can be constructed by SUT of CQF. Then, based on SUT of twodimensional CQF, a parameterized method is presented for constructing realvalued twodimensional orthogonal wavelet filters. Finally, nonseparable wavelet filters are derived. Conclusion remarks are given in Section 4.
2. Parametrization of OneDimensional Wavelet Filters
To construct onedimensional orthogonal scaling function we need construct sequences such that (see [4, 21] and many others) where denotes the Kronecker delta The sequence which satisfies (2.4) is called a onedimensional CQF, and the sequence , which satisfies (2.2), (2.3), (2.4) simultaneously, is called onedimensional orthogonal lowpass wavelet filter.
Definition 2.1. Let be a onedimensional CQF and an arbitrary orthogonal matrix. is called the shift unitary transform (SUT) of by if satisfies and is called the inverse shift unitary transform (ISUT) of by if satisfies where for any ,
The SUT and the ISUT of by are, respectively, denoted by By directly calculating, we have the following results.
Lemma 2.2. If is a onedimensional CQF, then and are onedimensional CQFs.
Lemma 2.3. If , then .
Therefore, for a onedimensional CQF, different onedimensional CQFs can be derived when we choose different orthogonal matrices . For a sequence , letting , we call the support of . If is a onedimensional CQF and the support of it is in , we call a simple onedimensional CQF.
Theorem 2.4. If is a onedimensional CQF, it can be constructed from a simple onedimensional CQF by a series of SUT.
Proof. It only needs to prove that by a series of ISUT of , we can get a simple onedimensional CQF.
Without loss of generality, we assume that the support of is in , and . We will prove this theorem by induction.
We see that the theorem is true for . Assume it is true for the case . We now prove it is true for the case . Suppose that is a onedimensional CQF and . We denote that
where . Recall that is a onedimensional CQF, that is, , which implies that . Therefore,
It follows that the support of is in . Suppose that , then . It follows that there exists , such that . Furthermore, if , then . By the hypothesis, the theorem is proved.
Theorem 2.4 shows that any onedimensional orthogonal lowpass wavelet filter can be constructed by a series of SUT. Suppose that an orthogonal scaling function satisfies (2.1). Then the sequence is a onedimensional CQF. By Theorem 2.4, we know that can be constructed by a simple onedimensional CQF and a series of orthogonal matrices. For a real number , we denote as the orthogonal matrix: Let be a simple onedimensional CQF such that For an arbitrary positive integer , choosing , we define as follows: In general, the support of is in . In other words, is a length filter. From the proof of Theorem 2.4, we know that the length filter can be constructed by at most times SUT of onedimensional CQF.
We can prove inductively the following theorems.
Theorem 2.5. Let , then
Theorem 2.6. Let then the sequence given in (2.14) is a onedimensional lowpass wavelet filter.
Theorem 2.6 provides a condition for choosing such that the onedimensional CQF given in (2.14) is a onedimensional orthogonal lowpass wavelet filter.
In addition, if the lowpass wavelet filter satisfies the Cohen's condition (see [4]), then the corresponding to in (2.2) is an orthogonal scaling function.
By Theorem 2.6, we know that a length onedimensional lowpass wavelet filter can be constructed by choosing such that condition (2.16) is satisfied. Therefore, any length onedimensional lowpass wavelet filter can be parameterized into a parameter family of wavelet system. In fact, we can give an explicit parametrization of any length filter: where .
Applications of onedimensional parameterized wavelets to compression are, for example, discussed in [22, 23]. Parameterizing all possible filter coefficients that correspond to compactly supported onedimensional orthonormal wavelets has been studied by several authors [6, 9, 24–26]. We provide explicit parametrization of any length filters which satisfy the necessary conditions for orthogonality in terms of SUT.
3. Construction of TwoDimensional Wavelet Filters
3.1. SUT of TwoDimensional CQF
To construct twodimensional orthogonal scaling function and its associated wavelets we need construct sequences and such that (see [10, 12, 13], etc.) where and .
The sequence , which satisfies (3.4), is called a twodimensional CQF. If satisfies (3.3), (3.4), and (3.5) simultaneously, we call it a twodimensional lowpass wavelet filter. The sequences , which satisfy (3.6) and (3.7), are called twodimensional highpass wavelet filters.
For an arbitrary realvalued sequence , we define and is called the support set of . If is finite, we call finite supported sequence. As aforementioned, the sequences we consider are realvalued and finite supported (FIR).
We note that any sequence can be split into 4 disjoint subsets
Definition 3.1. Let be an arbitrary orthogonal matrix, and let be an FIR. For and for all , we set Then , which is defined as follows: is called the twodimensional SUT of .
Lemma 3.2. If is a twodimensional CQF and is given by (3.11), then is also a twodimensional CQF.
Proof. By directly calculating, we can prove that satisfies the following equation: This completes the proof.
If the new twodimensional CQF is a lowpass wavelet filter, then it is worthwhile to construct the associated highpass wavelet filters. Now we provide a result of it.
Lemma 3.3. Suppose that is a twodimensional CQF, satisfy (3.6) and (3.7), is a twodimensional CQF obtained by SUT of , then , which are derived by SUT of , satisfy (3.6) and (3.7) simultaneously.
It can be proved by direct calculation, so we omit the proof.
Definition 3.4. (i) is called the SUT0 of if one chooses
in (3.11). This transform is denoted by .
(ii) is called the SUTT1 of if one chooses
in (3.11). This transform is denoted by . is called the SUTT2 of if one chooses
This transform is denoted by .
(iii) is called the SUT1 of if one chooses
in (3.11). This transform is denoted by . is called the SUT2 of if we choose
This transform is denoted by .
For a twodimensional CQF, when we choose different orthogonal matrices, many new twodimensional CQFs can be obtained. It is obvious that, after SUT0, the support of the new twodimensional CQF does not change. But it is different for SUTT1, SUTT2, SUT1, and SUT2. For example, the support of the twodimensional CQF is , and the support of the filter is , where and In general, for integers , if the support of is in , after SUT1 (or SUT2) the support of the new filter is in (or in ). If the support of is , we call a simple twodimensional filter.
We will adopt the following notations in the rest of this paper. For arbitrary , let be the FIR defined as follows: Furthermore, let be the filters as follows and let be orthogonal matrices such that respectively. Then is a simple twodimensional CQF, and satisfy (3.6) and (3.7).
From the special twodimensional CQF , by SUTT1, SUTT2, SUT1, SUT2, and the matrices , we can construct some new twodimensional CQFs.
3.2. TensorProduct Wavelet Filters
In this subsection, we will show that all tensorproduct wavelet filters can be constructed by SUTT1 and SUTT2.
A twodimensional lowpass wavelet filters is called tensorproduct wavelet filter if it satisfies the following equations: where , are onedimensional orthogonal lowpass wavelet filters.
Theorem 3.5. If is a tensorproduct lowpass wavelet filter, then, it can be constructed by SUTT1 and SUTT2 from .
Proof. , it follows from (3.24) that
Suppose that is a length onedimensional filter. Then there exist such that , and
We denote
Let be twodimensional filter such that
It follows that
Hence, can be constructed by SUTT1 of .
It can be proved inductively that can be constructed by a series of SUTT1 from , and can be constructed by a series of SUTT2 from , where denotes the filter
Therefore, any tensorproduct twodimensional lowpass wavelet filters can be constructed by SUTT1 and SUTT2 from .
3.3. TwoDimensional Wavelet Filters in Terms of SUT1 and SUT2
From now on, we give a method of construction of twodimensional orthogonal wavelet filters from a simple twodimensional CQF by SUT1 and SUT2.
For arbitrary positive integers and , choosing ,, is defined as where and Then is a twodimensional CQF, and its support is in . We are now in a position to draw some conditions on choosing , such that is a twodimensional lowpass wavelet filter.
Theorem 3.6. Let Then
Proof. It can be proved inductively. For the case , it is obviously true. Assume that it is true for the case ; now we prove that it is true for the case .
Let be the integer such that and . Suppose that
It follows that and . Let
it follows that
Let . Since , then
Therefore,
For , let
It follows from that
Therefore,
where
Namely, it is true for the case .
Similarly for the case . This completes the proof.
Now we provide the condition on choosing , such that is a twodimensional lowpass wavelet filter.
Theorem 3.7. If there exists integers and , such that then the sequence constructed in (3.31) is a twodimensional lowpass wavelet filter.
Proof. It follows from Theorem 3.6 that Therefore, satisfies conditions (3.3) and (3.5), then it is a twodimensional lowpass wavelet filter.
Corollary 3.8. If as given in (3.31) is a lowpass wavelet filter, then are the highpass filters associated with , where .
Remark 3.9. We can choose other orthogonal matrices than and such as where ; but conditions in Theorem 3.7 should be changed.
Example 3.10. We choose in (3.31). For , define as follows:
By choosing such that , we can get many twodimensional lowpass wavelet filters and their corresponding highpass wavelet filters. For instance, set , , , . By (3.31) and (3.47), we can get a nonseparable orthogonal lowpass wavelet filter (see Table 1) and its associated highpass 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.




(a)
(b)
Remark 3.11. By Section 3.2, we know that any twodimensional tensorproduct 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 twodimensional wavelet filters in terms of SUT of twodimensional 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 onedimensional CQF, any onedimensional orthogonal wavelet filters with dilation factor 2 can be given in explicit expression. The SUT of twodimensional CQF is applied to the construction of twodimensional orthogonal wavelet filters, and a parametrization method is presented. The selection of the parameters is not restricted by any implicit condition. Tensorproduct 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.
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Copyright
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.