Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 624957, 11 pages

http://dx.doi.org/10.1155/2013/624957

Research Article

Construction of Bivariate Nonseparable Compactly Supported Orthogonal Wavelets

Jinsong Leng,¹ Tingzhu Huang,¹ and Carlo Cattani²

¹School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China

²Department of Mathematics, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano, Italy

Received 7 January 2013; Accepted 24 January 2013

Academic Editor: Sheng-Yong Chen

Copyright © 2013 Jinsong Leng et al. 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.

Abstract.

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 .

1. Introduction

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 [3]. 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 [13] 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 [18] to construct nonseparable wavelets using the dilation matrix with determinant equal to . He and Lai [16] constructed many examples of nonseparable orthogonal wavelets. Lai and Roach [19] 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., [9]), 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 [13] 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.

2. Preliminaries

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 [26] 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 [13] 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 [13]). The high-pass filter coefficients are given by

where , and are low-pass filter coefficients.

Lai [18] 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 [18]).

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 [27]. 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 [13].

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 [13].

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 [27]. Hence satisfies Cohen’s criterion (18).

3.2. Accuracy

We will use the same method as given in [13] 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 [14], 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 [13], 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 [13].

3.3. Smoothness

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 [13].

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 [2], 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 [13], (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).

3.4. Example

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 [27]):

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 [13]). One of them is given as follows:

where the coefficients are displayed in Table 1.

Table 1: The coefficients of the polynomials and of the example given in Section 3.4

0
0.007903570868, 0.036969146934

1
−0.013689386305, −0.064032440902

2
−0.138084015734, −0.056780853066

3
0.121182418797, 0.137970734671

4
0.549926093940, 0.002654265329

5
0.346172096397, −0.083844146934

6
0.080254350926, 0.017157440802

7
0.046334871111, 0.009905853066

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.

Table 2: The coefficients of the low-pass filter of the example given in Section 3.4

0
0.002629285291, 0.006377324227, 0.015933220981, 0.028763021603

1
−0.004554055712, −0.011045849577, −0.027597148268, −0.049819014795

2
−0.045936485868, −0.111418819867, −0.082391050025, −0.027173124419

3
0.040313822269, 0.097781064805, 0.101627868651, 0.094966452370

4
0.182944217753, 0.443730695949, 0.253781634345, −0.072104378716

5
0.115161262724, 0.279323325343, 0.131304280272, −0.114390178100

6
0.026698259299, 0.064756554347, 0.042614945757, 0.003008971526

7
0.015414247193, 0.037387214084, 0.024603750404, 0.001737230521

0
0.016325987784, −0.004712828742, −0.003158691008, 0.001302285961

1
−0.028277440326, 0.008162858829, 0.005471013312, −0.002255625450

2
−0.014314162669, 0.002801853046, 0.004851428418, −0.002000178904

3
0.053095656592, −0.014358762363, −0.011788395329, 0.004860197371

4
−0.045765749098, 0.019013502031, −0.000226783811, 0.000093499925

5
−0.068135421758, 0.023514243110, 0.007163750722, −0.002953518393

6
0.001033287041, 0.000510619607, −0.001465953599, 0.000604393018

7
0.000596568551, 0.000294806368, −0.000846368705, 0.000348946471

(a)

(b)

Figure 1: Plot of the real (a) and imaginary (b) parts of the bivariate compactly supported low-pass filter (74).

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.

Figure 2: The scaling function generated by the low-pass filter (74).

4. Conclusion

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.

Acknowledgments

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).

References

C. K. Chui, An Introduction to Wavelets, vol. 1 of Wavelet Analysis and Its Applications, Academic Press, Boston, Mass, USA, 1992.

I. Daubechies, Ten Lectures on Wavelets, vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1992.

A. Said and W. A. Pearlman, “A new fast and efficient image codec based on set partitioning in hierarchical trees,” IEEE Transactions on Circuits and Systems for Video Technology, vol. 6, no. 3, pp. 243–250, 1996.

C. Cattani and J. Rushchitsky, Wavelet and Wave Analysis as Applied to Materials with Micro or Nanostructure, vol. 74 of Series on Advances in Mathematics for Applied Sciences, World Scientific, Singapore, 2007.

S. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way, Academic Press, Amsterdam, The Netherlands, 3rd edition, 2008.

G. Strang and T. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press, Wellesley, Mass, USA, 1996.

M. Vetterli and J. Kovacevic, Wavelets and Subband Coding, Prentice Hall, New York, NY, USA, 1995.

C. Cattani and L. M. S. Ruiz, “Discrete differential operators in multidimensional Haar wavelet spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 2004, no. 44, pp. 2347–2355, 2004.

A. Cohen and I. Daubechies, “Nonseparable bidimensional wavelet bases,” Revista Matemática Iberoamericana, vol. 9, no. 1, pp. 51–137, 1993.

P. Wojtaszczyk, A Mathematical Introduction to Wavelets, vol. 37 of London Mathematical Society Student Texts, Cambridge University Press, Cambridge, UK, 1997.

A. Ayache, “Construction of non-separable dyadic compactly supported orthonormal wavelet bases for of arbitrarily high regularity,” Revista Matemática Iberoamericana, vol. 15, no. 1, pp. 37–58, 1999.

A. Ayache, “Some methods for constructing nonseparable, orthonormal, compactly supported wavelet bases,” Applied and Computational Harmonic Analysis, vol. 10, no. 1, pp. 99–111, 2001.

E. Belogay and Y. Wang, “Arbitrarily smooth orthogonal nonseparable wavelets in ,” SIAM Journal on Mathematical Analysis, vol. 30, no. 3, pp. 678–697, 1999.

C. Cabrelli, C. Heil, and U. Molter, “Accuracy of lattice translates of several multidimensional refinable functions,” Journal of Approximation Theory, vol. 95, no. 1, pp. 5–52, 1998.

B. Han, “Symmetric multivariate orthogonal refinable functions,” Applied and Computational Harmonic Analysis, vol. 17, no. 3, pp. 277–292, 2004.

W. He and M. J. Lai, “Examples of bivariate nonseparable compactly supported orthonormal continuous wavelets,” IEEE Transactions on Image Processing, vol. 9, no. 5, pp. 949–953, 2000.

A. Karoui, “A note on the design of nonseparable orthonormal wavelet bases of ,” Applied Mathematics Letters, vol. 18, no. 3, pp. 293–298, 2005.

M. J. Lai, “Methods for constructing nonseparable compactly supported orthonormal wavelets,” in Wavelet Analysis: Twenty Year's Development, D. X. Zhou, Ed., vol. 1, pp. 231–251, World Scientific, Singapore, 2002.

M. J. Lai and D. W. Roach, “Nonseparable symmetric wavelets with short support,” in 7th Wavelet Applications in Signal and Image Processing, vol. 3813 of Proceedings of SPIE, pp. 132–146, 1999.

J. S. Leng, T. Z. Huang, and Y. D. Fu, “Construction of bivariate nonseparable compactly supported biorthogonal wavelets,” in Proceedings of the International Conference on Machine Learning and Cybernetics, pp. 3625–3629, Kunming, China, July 2008.

J. S. Leng, T. Z. Huang, Y. D. Fu, and C. H. Lai, “A method for constructing trivariate nonseparable compactly supported orthogonal wavelets,” International Journal of Computer Mathematics, vol. 86, no. 7, pp. 1264–1273, 2009.

J. S. Leng, T. Z. Huang, Y. F. Jing, and W. Jiang, “A study on conjugate quadrature filters,” EURASIP Journal on Advances in Signal Processing, vol. 2011, Article ID 231754, 2011.

R. L. Long, Multi-Dimensional Wavelet Analysis, Springer, Beijing,China, 1995.

D. Stanhill and Y. Y. Zeevi, “Two dimensional orthogonal wavelets with vanishing moments,” IEEE Transactions on Signal Processing, vol. 44, no. 10, pp. 2579–2590, 1996.

D. Stanhill and Y. Y. Zeevi, “Two dimensional orthogonal filter banks and wavelets with linear phase,” IEEE Transactions on Signal Processing, vol. 46, no. 1, pp. 183–190, 1998.

L. F. Villemoes, “Continuity of nonseparable quincunx wavelets,” Applied and Computational Harmonic Analysis, vol. 1, no. 2, pp. 180–187, 1994.

I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Communications on Pure and Applied Mathematics, vol. 41, no. 7, pp. 909–996, 1988.