Abstract
A construction approach for the 3-band tight wavelet frames by factorization of paraunitary matrix is developed. Several necessary constraints on the filter lengths and symmetric features of wavelet frames are investigated starting at the constructed paraunitary matrix. The matrix is a symmetric extension of the polyphase matrix corresponding to 3-band tight wavelet frames. Further, the parameterizations of 3-band tight wavelet frames with filter lengths are established. Examples of framelets with symmetry/antisymmetry and Sobolev exponent are computed by appropriately choosing the parameters in the scheme.
1. Introduction
In the theory and applications of wavelets and wavelet frames, certain properties are always desirable. It is well known that symmetry and high vanishing moments are very important features of all wavelets. Symmetry, which is also called linear phase in the language of engineering, is claimed to improve the rate-distortion performance in image compression [1, 2]. On the one hand, parameterizations of FIR systems are of fundamental importance to the design of filters with the desired properties [1, 3]. On the other hand, the advantages of MRA-based tight wavelet frames and their promising features in applications have attracted a great deal of interest and effort in recent years to extensively study them (e.g., [4–15]). The main tools for construction and characterization of wavelet frames are the unitary extension principle (UEP) [16] and its versions generalized such as OEP and MEP [17]. They give sufficient conditions for constructing MRA-based tight and dual wavelet frames. Many authors have worked on the design of wavelet frames with good properties. Most deals with 2-band wavelet frames systems, and a few authors have studied -band framelets [8]. -band wavelets have advantages over dyadic wavelets in some aspects. For example, it enables a finer frequency partitioning and can provide a more compact representation of signals [18–21].
This paper deals with the construction of 3-band tight wavelet frames filters with prescribed properties using factorization and parameterizations of the paraunitary matrices. The parameter space describing 3-band wavelets is much richer than that in 2-band case; thus, it has greater freedom and flexibility. Concretely, with the describing of unitary extension principle (UEP) in the polyphase representation, we firstly construct a paraunitary matrix based on the polyphase matrix corresponding to compactly supported wavelet frames with the least number generators. Further, we establish necessary constraints on the filter lengths and symmetric features of wavelet frames. Then, we investigate the parameterizations of 3-band tight wavelet frames with filter lengths. Finally, examples of 3-band wavelet frames with symmetry/antisymmetry and good smoothness are given by applying the proposed scheme.
Throughout this paper, let and denote the sets of all natural numbers and integers, respectively. Let , , and denote conjugate transpose, transpose and the trace of , respectively. Let , ; denote the matrix with elements , for convenience, we omit the subscript when . Let denotes the identity matrix and denote the exchange matrix with ones on the antidiagonal. For , we use denote the Sobolev space consisting of all functions with . In this paper, we only consider compactly supported wavelet frame and causal sequence with real finite impulse response (FIR), such sequences can be identified with Laurent polynomial defined by , where and are the smallest and largest indices that is nonzero, respectively. Assumpsit: and . We use leng to denote the filter length of .
2. UEP of Tight Framelets in Terms of Polyphase Representation
In this section, we introduce some notation and state needed results that will be used later in this paper. Let refinable functions with dilation factor 3 generate a multiresolution analysis (MRA) of and . Then for real-value sequence , , satisfy or equivalently where or rewritten , . is called a refinement mask or the low filter, and , are called wavelet masks or the high pass filters of the system, respectively. For notational convenience, refinement mask together with wavelet masks is also called combined MRA masks in [7].
For given , define the wavelet system as , where .
Definition 2.1. The system is called an MRA-based 3-band tight wavelet frames of if (1),
(2) is tight wavelet frames, that is,
holds for all . Furthermore, elements , are said to be framelets.
If we introduce the following two block matrices: modulation matrix
and polyphase matrix
where
then we describe the UEP for 3-band tight wavelet frames in terms of modulation matrix and polyphase matrix as follows.
Lemma 2.2. Let be a compactly supported refinable function generated by finitely supported refinement mask with . Then for combined MRA masks , the system , where defined by (2.1) or (2.2) forms an MRA-based 3-band tight wavelet frame of provided one of the following hold for . (a)Modulation matrix is paraunitary, that is .(b) Polyphase matrix is paraunitary, that is .
Symmetry and high vanishing moments are very important features of all wavelets.
Definition 2.3. A casual FIR filter is called symmetric/antisymmetric, if , , where is called the center of .
Lemma 2.4. Suppose that the mask is symmetric/antisymmetric, then defined by (2.1) is symmetric/antisymmetric about , respectively, that is, , where .
The vanishing moments of wavelets is related to the order of sum rules.
Definition 2.5. The refinement mask has the sum rules of order , if
or equivalently
Lemma 2.6. Suppose that the wavelet mask satisfies , , then defined by (2.1) has vanishing moments of order , respectively, that is, , . Where .
3. Symmetry Transform
In this section, we will firstly construct a paraunitary matrix based on the polyphase matrix corresponding to 3-band compactly supported wavelet frames. Further, we will establish necessary constraints on the filter lengths and symmetric features of wavelet frames.
3.1. Construction of Paraunitary Matrix Based on the Polyphase Matrix
From now on, we only consider the least number wavelet frames with and for the sake of convenience, we rewrit refinement mask and wavelet frame mask associated with and , , respectively, that is, , , , , where .
Suppose casual FIR filter , , , are symmetric/antisymmetric and , , , , defined by (2.4). For , define where Then can be seen as a symmetry paraunitary extension of polyphase matrices with , and have the following theorem and further investigation.
Lemma 3.1. Let , , and are defined in (2.4), (2.5), and (3.1). Then the following three statements are equivalent.(a);(b);(c).
From the foregoing discussion Further, if , , , are causal, then for a sufficiently large is also causal. A paraunitary and causal FIR matrix can be factorized into the products of [1, 3], where . So far, the construction for FIR filters , , , converted the design for paraunitary matrix .
3.2. Investigation on Filter Lengths and Symmetry Features
The main content of this section is to investigate the constraint conditions of filter lengths and symmetric features of wavelet frames in order to discuss the method for construction framelets with desired properties.
Theorem 3.2. Let , , , are symmetric/antisymmetric nonzero FIR filters with the centers of symmetry , , , , respectively, where , , , and . If is paraunitary, then(1).(2), have the same parity.
Proof. By the conditions of , we have leng, leng, leng, leng, where , , , . From the paraunitariness of , we have , leng = leng. Thus is even.
From the paraunitariness of , we obtain
or
where or . From the symmetric of , , , , we have
Further,
Thus
Since is even, hence
Obviously, is even when is even. Assume that is even when is odd, then we have , which is contradiction since is nonzero filter.
Without loss of generality, we give two assumptions of the casual filters : (1) leng leng leng( leng(mod3); (2) lenglengleng leng.Then we have the following Theorem.
Theorem 3.3. Let are the causal filters and satisfy and , . Suppose is the center of and , , , . If is a paraunitary matrix, then (1), , , and are even; (2), ; (3) defined by (3.1) with is causal, paraunitary, and satisfies
Proof. By the symmetry of , , , , we have
Take them into defined by (3.2), we have
Similarly, for , , defined by (3.3)–(3.5), we also have
Thus, defined by (3.1) satisfies with
From the paraunitariness of , we obtain leng = leng. Assume that
By symmetric of , we have , which together with (3.17) leads to leng; thus, leng is even. Note that and have the same parity by Theorem 3.2, hence is even, is even. One can prove similarly that , and are all even.
Let . Then defined by (3.2)–(3.5) with satisfy
Note that
Thus, , , , are casual, is casual, paraunitary, and satisfies (3.14).
Set and in (3.14), then we have
which imply (2.2).
With similar arguments we can prove the others case and present the following two theorems as follows.
Theorem 3.4. Let are the causal filters and satisfy and , . Suppose is the center of and , , , , . If is a paraunitary matrix, then (1) and are even. (2), . (3) defined by (3.1) with is causal, paraunitary and satisfies
Theorem 3.5. Let are the causal filters and satisfy and , . Suppose is the center of and , , , . If is a paraunitary matrix, then (1), , , and are even; (2), ; (3) defined by (3.1) with is causal, paraunitary and satisfies
So far, we have completed the further extension of the our previous work in [22], that is, we presented the several necessary constraints on the filter lengths and symmetric features of wavelet frames. By using these properties, defined by (3.1) can be divided into several categories according to the different length of masks. Then we can improve the condition of in the following work and prepare for the parameterizations. This paper is devoted to give the parameterizations of 3-band tight wavelet frames with filter lengths.
4. The Parameterizations for Combined MRA Masks with Filter Lengths
For simplicity we give the following two assumptions: (1) lengleng+1, lengleng; (2), .
4.1. The Case of
For with in (3.1), denote , then (3.24) can be converted to While imply that(i) The first case: one of is and the other two are ;(ii) The second case: one of is and the other two are .
Lemma 4.1. For in (4.1), let where , then satisfies if and only if satisfies
(i) For the first case, without loss of generality, suppose , (4.4) is converted to To give the parameterizations of satisfying (4.5), one defines One can check that is paraunitary and satisfies
Lemma 4.2. For in (4.3), define Then satisfies
Lemma 4.3. Let a paraunitary matrix defined by (4.8), then for , have the form of where .
Proof. Let , (4.9) is converted to If set matrix , , then we can obtain Assume that , , then (4.10) is obtained immediately.
Remark 4.4. For convenience, we only choose the following form of in the following:
Lemma 4.5. Let a paraunitary matrix satisfies (4.8), then for , have the form of where , .
Proof. When , (4.9) is converted to
Suppose , here , , , are all matrices. Let then, from the paraunitariness of , we have
where , , , are all not zeros or , , , are all not zeros. Now suppose , , , are all not zeros, then , .
Similarly, and are given as follows:
Equation (4.14) is obtained.
Theorem 4.6. A causal paraunitary matrix filter defined by (4.8) for some if and only if it can be factorized in the form of where is defined by (4.13) with and , are defined by (4.14) with , .
Proof. If can be factorized as (4.18), then it is a causal paraunitary filter and satisfies (4.9). Conversely, we set a paraunitary matrix satisfies with (4.18), where . If , it is easy to get that is . For , define
where is defined by (4.14). Now we prove that there exists , such that defined by (4.19) is causal. From , we only need to prove that there exists , such that
Let us start by discussing the case . Then (4.20)–(4.22) hold. By the symmetry of , , . By the paraunitariness of , . Then we have , . For simplicity, we only consider the case in the following. Let , , . Then for , , (4.23) and (4.24) hold. Let , by the paraunitariness of , . Then we have . It is clear that since . Then (4.25) holds for , .
If , we have and . Let , then (4.20)–(4.22) hold for , . Similarly, when , (4.23), and (4.24) hold for , . Let , from the paraunitariness of , we have . Thus, . It is clear that since . So (4.25) holds for , .
So is causal and can be factorized in the form of . Thus, can be factorized in the form of by induction assumption. The proof of Theorem 4.6 is complete.
Therefore, we have the following corollary.
Corollary 4.7. Let , , , is symmetric or antisymmetric. If is a paraunitary matrix, then , , , can be factorized as
for the case of , ;
for the case of , , , where ;
for the case of , , where .
Where is defined by (4.13) with and are defined by (4.14) with .
(ii) For the second case, without loss of generality, suppose , , , then defined by (4.8) satisfies Let in (4.29), then one can get an orthogonal matrix as follows: Then one has the following results.
Theorem 4.8. Let is a causal paraunitary matrix filter and satisfies (4.29) for some , then cannot be factorized in the form of for defined by (4.30) with and , defined by (4.14) with , .
Proof. Suppose can be factorized in the form of (4.31). Then when , we have when , we have However, (4.33) is in contradiction with (4.32).
4.2. The Case of
We have the following results.
Theorem 4.9. is a causal paraunitary matrix filter defined by (4.8) for some if and only if it can be factorized in the form of
where is defined by the following with ,
and are defined by (4.14) with .
Corollary 4.10. Let , , , is symmetric or antisymmetric. If is a paraunitary matrix, then , , , can be factorized as for the case , , ; for the case of , . Where is defined by (4.35) with and are defined by (4.14) with .
Remark 4.11. In the case of , , we cannot construct a paraunitary matrix . The reason is similarly as Theorem 4.8 when , .
4.3. The Case of
In this situation, we cannot also construct the corresponding paraunitary matrix.
5. Examples
In this section, we will construction some examples of 3-band wavelet frames with symmetry/antisymmetry by applying the parameterizations of masks , , , provided in Corollarys 4.7 and 4.10.
Example 5.1. Let , , , be the filters given by (4.36) with . Then For , , , , we have , , , . Let has sum rules of order 2, such as , then we can get that , . Thus The resulting symmetric refinable function . See Figure 1 for the graphs of , , , and .
(a)
(b)
(c)
(d)
Example 5.2. Let , , , be the filters given by (4.26) with . For , , , let , , , , , , then we have
Let has sum rules of order 2, such as , then we can get that , , , , , .
Thus
The resulting symmetric refinable function . See Figure 2 for the graphs of , , , and .
(a)
(b)
(c)
(d)
Acknowledgments
The authors would like to thank the support of the National Natural Science Foundation of China (Grant no. 61272028). The authors also appreciate the support of the Fundamental Research Funds for the Central Universities (no. ZZ1229), National Undergraduate Training Programs for Innovation and Entrepreneurship (no. 201210010077) and Science and Technology Innovation Foundation for the College Students (no. pt2012064).