Information and Modeling in Complexity 2013View this Special Issue
Construction of Bivariate Nonseparable Compactly Supported Orthogonal Wavelets
A method for constructing bivariate nonseparable compactly supported orthogonal scaling functions, and the corresponding wavelets, using the dilation matrix , is given. The accuracy and smoothness of the scaling functions are studied, thus showing that they have the same accuracy order as the univariate Daubechies low-pass filter , to be used in our method. There follows that the wavelets can be made arbitrarily smooth by properly choosing the accuracy parameter .
Wavelet analysis (see [1, 2]) has become a powerful tool in neural networks, harmonic analysis, numerical analysis, and signal processing, especially in the area of image compression . The wavelet transform is a simple and practical mathematical tool that cuts up data or functions into different frequency components and then studies each component with a resolution matched to its scale. The main feature of the wavelet transform is to hierarchically decompose general functions, as a signal or a process, into a set of approximation functions with different scales [1, 2, 4, 5].
Wavelet theory is closely related to subband coding, and it provides a functional space structure for subband coding, often leading to a better understanding of signals.
Univariate wavelets have found successful applications in signal processing [5–7]. However, in order to apply wavelet methods to data processing and to analyze multidimensional signals, such as image processing, we have to construct multivariate wavelets. The most commonly used method to define multivariate wavelets is the tensor product of univariate wavelets [4, 8–10]. This construction leads to a separable wavelet function having, as a major drawback, the horizontal and vertical (obliged) directions, while usually in imaging process the information contents spread along all directions, in a nonisotropic way. Much efforts have been spent for constructing multivariate nonseparable compactly supported orthogonal wavelets (see, e.g., [11–26]); however, the generalization to higher dimensions is not a simple task, so that a general simple methodology is still missing. However, it is possible to characterize the construction of bivariate scaling functions by the two scale difference equation, which in 1 dimension is and in 2 dimensions depends on the coefficients of a suitable second-order dilation matrix . Usually [9, 11–13] the determinant of this matrix is 2. For example, Ayache [11, 12] proposed two methods for constructing bivariate nonseparable compactly supported orthonormal wavelets. The wavelets constructed using one of the methods are called semiseparable wavelets. Belogay and Wang  constructed bivariate nonseparable compactly supported orthonormal wavelets using the dilation matrix . Belogay and Wang gave the sufficient condition for the low-pass filters constructed by their method to generate nonseparable compactly supported orthogonal accurate scaling function with arbitrarily high smoothness. However, for proving the accuracy these authors used the matrix . Moreover, their method has been modified by Lai  to construct nonseparable wavelets using the dilation matrix with determinant equal to . He and Lai  constructed many examples of nonseparable orthogonal wavelets. Lai and Roach  provided a method to construct bivariate nonseparable compactly supported wavelets by using the method of symmetry. Stanhill and Zeevi [24, 25] provided other methods for constructing bivariate nonseparable orthogonal wavelets.
By combining the Belogay and Wang method, the Lai method, and the Ayache method, we present another method for constructing bivariate nonseparable orthogonal wavelets using the dilation matrix . Our method holds true also for dilation matrix of the form being the identity matrix, so that It follows that the two-scale difference equation simplifies into which is more alike the corresponding equation in 1 dimension. The minor drawback is that for the multivariate scaling function there exist corresponding wavelets (see, e.g., ), so that for a matrix with determinant 4 there exist, together with a scaling function, 3 wavelet functions (instead of 1 wavelet when ). However, due to the special form of the dilation matrix, we can take into account the results of [11–13] and show that the scaling function constructed by our method has the same accuracy as the univariate Daubechies low-pass filter to be used in our method. Following  we will also show that the smoothness of our bivariate scaling function can be manipulated by properly choosing the accuracy parameter .
The paper is organized as follows. In Section 2, we introduce some notations and briefly recall some basic elements of the theory of bivariate multiresolution analysis and bivariate orthogonal wavelets. In Section 3, we provide a method for constructing a class of bivariate nonseparable compactly supported orthogonal filter banks. Then we study the accuracy and the smoothness of the scaling function constructed by our method. Finally, we give an example of the bivariate nonseparable compactly supported scaling function.
Let and be the sets of all integers and real numbers, respectively. Denote The Fourier transform of is defined by where , and is the dot product.
For , their inner product is defined by which is explicitly
A function is called orthogonal if the set of its translate is orthogonal; that is, where and is the Kronecker notation defined by if and if , which is explicitly
The whole construction of wavelets is based on the multiresolution analysis (MRA) and its corresponding axioms, which can be easily extended to the bivariate multiresolution analysis as follows. Let satisfy the following refinement equation: where is a real constant sequence.
We assume that for only finitely many (ensuring that has compact support).
Define a closed subspace by
Definition 1. One says that in (12) generates a bivariate multiresolution analysis (MRA) of if the sequence defined by (13) satisfies(i); (ii)is dense in and , where 0 is the zero function; (iii); (iv)is an orthogonal basis of .The function is called an orthogonal scaling function of the bivariate multiresolution analysis .
By taking the Fourier transform for both sides of (12), we have where is a Laurent polynomial satisfying .
The orthogonality condition (10) implies that The function is called a low-pass filter (refinement mask). By (14) and , we get so that the scaling function can be defined by a suitable choice of the low-pass filter.
Let ; the filter satisfies Cohen’s criterion  if there exists a compact fundamental domain of the lattice with the property
Let us note that (16) is only a necessary condition for to be orthogonal. The condition (16) becomes also a sufficient condition if the filter satisfies Cohen’s criterion (18).
Let us now assume that there exist three closed subspaces , such that where denotes orthogonal direct sum of spaces. We say that the compactly supported functions are orthogonal wavelets if form orthogonal bases of ; that is, where and .
If there exist three Laurent polynomials , such that is unitary, that is, , then the Fourier transforms of the functions , can be given by and the three Laurent polynomials , are called the high-pass filters (wavelet masks). The whole set of functions is called a filter bank.
By (17) and (22), the construction of bivariate compactly supported orthogonal scaling function and the corresponding wavelets is reduced to the construction of filter bank which satisfies (16), (18), and (21).
3. Construction of Bivariate Nonseparable Orthogonal Wavelets
3.1. Construction of the Filter Banks
According to Section 2, for constructing bivariate orthogonal scaling function and the corresponding wavelets, we mainly need to define the bivariate orthogonal filter bank satisfying (16), (18), and (21).
Let us shortly recall the Belogay and Wang method, the Lai method, and the Ayache method for constructing bivariate nonseparable compactly supported orthogonal wavelets and their corresponding filter banks.
Belogay and Wang  constructed the bivariate nonseparable orthogonal scaling functions using the dilation matrix with special coefficients: the coefficients are aligned along two adjacent rows. The corresponding low-pass filters have the form where are two univariate polynomials of satisfying some conditions (see ). The high-pass filter coefficients are given by where , and are low-pass filter coefficients.
Lai  modified the Belogay and Wang method to construct nonseparable wavelets using the dilation matrix . According to the Lai method, their filter bank has the form where are two univariate polynomials of satisfying some conditions, , and (see ).
Ayache [11, 12] proposed two methods for constructing bivariate compactly supported orthogonal wavelets using the dilation matrix . Let be two univariate trigonometric polynomials, such that Then according to one of the two methods, the nonseparable filter banks have the form where are two given univariate filter banks, , and .
By combining the previous three methods, we propose, in the following, a method based on a suitable choice of the univariate filter bank, together with the basic ideas of Belogay-Wang approach. In particular, we use the univariate Daubechies filter bank to define the bivariate nonseparable filter bank as follows.
Let be a nonnegative integer, and let be the univariate Daubechies filter bank [2, 27] with accuracy ; that is, where and . Let be two univariate polynomials of with real coefficients, and let . We define four bivariate polynomials of , which are the forms of the filter banks of this paper, as follows:
Let be a nonnegative integer, and let be an odd integer with . Now, we define the polynomials and of as follows: where , , and are given by where , are two arbitrary polynomials and satisfies (37).
The following condition (to be used in the following) is given for the minimal degree of and :
Remark 2. Obviously, the polynomial defined by (36) satisfies This equation was solved by Daubechies in . If (38) holds, then .
Lemma 3. The polynomial given by (30), where is the univariate Daubechies filer defined by (28), satisfies (16) if and only if the polynomials , in (30) satisfy
Proof. Note that
If (40) and (41) hold, we have
Hence (16) holds.
Conversely, if (16) holds, we have for all and . Hence (40) and (41) hold.
Remark 4. Condition (41) means that contains no even powers of . Therefore, must be odd if .
Lemma 5. The following conditions are equivalent: (i)the univariate real polynomials and satisfy condition (41) and ;(ii)there exist an odd integer number and real polynomials , , and , such that , , and .
Proof. See the proof of Lemmas 3.2 and 3.3 of .
Lemma 6. Let the polynomials and be given by (34)–(38) for ; then if and only if .
Proof. See the proof of Lemma 4.1 and Lemma 4.7 of .
By collecting the previous results and definition we can easily show the main result of this paper that is as follows.
Theorem 7 (existence). Let and be, respectively, a nonnegative integer and an odd integer number with ; let the polynomials and be given by (34)–(37); let the filter be given by (28); let the polynomials , , , and be defined by (30)–(33). Then is a bivariate nonseparable filter bank satisfying (16) and (21). If, in addition, the polynomials and satisfy condition (38), then satisfies Cohen’s criterion (18), and the filter bank can be used, according to (17), (22), to define bivariate nonseparable compactly supported orthogonal wavelets.
Proof. According to Lemmas 3 and 5 by letting
in order to prove that satisfies (16), we need only to show that , satisfy (40). Taking into account (34) and (35), we have
Hence, (40) follows from (39).
By a simple calculation, we can also get that satisfies (21). In fact, let us assume that where and are univariate low-pass filters. Then we have This leads to the contradiction that Hence is nonseparable.
Finally, suppose that and satisfy (34)–(38). Let , then is a compact fundamental domain of the lattice . When and , it is , so that we have for all and by using Lemma 6 and the result of . Hence satisfies Cohen’s criterion (18).
We will use the same method as given in  to study the accuracy of our scaling function. Let recall its definition and how the accuracy of the scaling function implies some restrictions on the low-pass filter.
According to the result of , an orthogonal scaling function with the dilation matrix has accuracy if and only if its low-pass filter satisfies the following accuracy condition: for all , with . Therefore, when satisfies (52), we say that it has accuracy .
Lemma 8. Let the polynomials and be given by (28) and (30), respectively; then satisfies (52) if and only if the polynomials in (30) defined by (34) have the form where , are polynomials with real coefficients satisfying
Proof. It is easy to check that
for all and .
If , satisfy (53) and (54), it is not difficult to show that satisfies also (52). On the other hand, if satisfies (52) and noting that , we have So that is a root of multiplicity of . Hence have the form of (53), and yields (54).
Lemma 9. Let the polynomials and be given by (28) and (30), respectively, and assume also that and in (30) are polynomials with and . Then satisfies (41) with accuracy if and only if there exist an odd integer number and real polynomials , , and , such that and satisfy (34) and (37).
Proof. If there exist an odd integer and real polynomials , and such that and satisfy (34), and (37), then satisfies (41) and has accuracy by Lemmas 5 and 8 by letting
Now, suppose that satisfies (41) with accuracy . By Lemma 8, is a root of multiplicity of . Since by (41), is another root of multiplicity of . But , so is a root of multiplicity of . Hence, we have
By substituting (58) in (41), we have
Then, by applying Lemma 5 to and , we get that there exist an odd integer and real polynomials , , and , such that (34) hold.
Since , and therefore , we can normalize and , so that . And follows from condition (54). Hence and satisfy (34), and (37).
Theorem 10 (accuracy). Let the polynomials , , and be given by (34)–(38) and (30) with according to (28). Then is a low-pass filter which can generate bivariate nonseparable compactly supported orthogonal scaling function with accuracy .
Proof. This theorem follows from Lemmas 8, 9, and Theorem 7.
Remark 11. (i) According to the proof of Theorem 2.1 of , if , satisfy (40), the polynomials and in Lemma 9 must satisfy (35) and (36).
(ii) The existence of and satisfying (35)–(38) is ensured by Corollary 4.2 of .
In this section, we measure the smoothness of the scaling function generated by by its Hölder exponent (see [13, 23]). Let , where is a nonnegative integer and . For nonnegative integers , and , denotes the partial derivative of ; that is, If there exists a constant , such that we say that . It is well known that if the Fourier transform of satisfies for some constants , , then (see [13, 23]).
Let us assume that the polynomials and satisfying (34)–(38) are and let
Define a polynomial as follows:
Lemma 12. Let be given by (64). Then there exist constants and , such that where and . Furthermore,it is
Proof. See the proof of Lemma 4.1, 4.3–4.5 of .
Theorem 13 (smoothness). Let the polynomials , , and be given by (28) and (34)–(38), respectively. If is the bivariate scaling function generated by the low-pass filter according to (30) and , then .
Proof. We only need to prove that the Fourier transform of satisfies (62). In fact, by applying Lemmas 7.1.1–7.1.8 of , we have for the same as Lemma 12, and Thus, according to (63), (64), (68), and (69) and Lemma 12, we get Hence by using (17) it is shown that (62) holds for .
Remark 14. (i) The wavelets have the same smoothness as the corresponding scaling function. So we only need to study the smoothness of the scaling function.
(ii) As the remark of paper , (67) mean that the smoothness of the bivariate orthogonal wavelets improves asymptotically by 0.2075 when is incremented by 1. Hence, (67) guarantees the existence of bivariate orthogonal wavelets of any desired smoothness. In particular, if , then , if , then , and if , then .
According to the results of this section, we can summarize the following procedure for constructing the filter banks.
Step 1. Choose a nonnegative integer , and let .
Step 2. Use some spectral factorization method to make a list of all polynomials and that satisfy (35)–(38). Substitute , and in (34) to get and .
Step 3. Construct the low-pass filter by substituting , and univariate Daubechies low-pass filter in (30).
Step 4. Construct the high-pass filters by substituting , and univariate Daubechies filters and in (31)–(33).
Finally, we give an example of bivariate nonseparable compactly supported orthogonal low-pass filter by using our method.
Let . The following polynomial is the univariate Daubechies low-pass filter with accuracy 3 (see ): where
If (38) holds, then there are four polynomials satisfying (35), (37) and two polynomials satisfying (36), (37). Hence, one can get 8 kinds of and satisfying (34)–(38) (see ). One of them is given as follows: where the coefficients are displayed in Table 1.
Then, by using (30), we can get a bivariate nonseparable compactly supported orthogonal low-pass filter (Figure 1) given as follows: where the coefficients are displayed in Table 2.
For instance, by explicit computation and the approximation of coefficients the first few terms of the filter look like By the inverse Fourier transform we finally obtain the scaling function, so that we have a sequence in the following form:
The scaling function generated by the low-pass filter (Figure 2) has accuracy 3. By (31)–(33), one can easily obtain the corresponding high-pass filters. We omit it here.
We gave a method for constructing bivariate nonseparable compactly supported orthogonal wavelets by combining three methods using the dilation matrix . We studied the accuracy of the scaling functions and the smoothness of the wavelets constructed by this method. We proved that the scaling functions have the same accuracy order as the univariate Daubechies low-pass filter used in this method and the wavelets can be made arbitrarily smooth by choosing the accuracy order.
The authors wish to thank the anonymous reviewers for their valuable comments and suggestions that have improved the presentation of this paper. This work was supported by the National Natural Science Foundation of China (11271001, 61170311), 973 Program (2013CB329404), and Sichuan Province Science and Technology Research Project (12ZC1802).
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