Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2011, Article ID 593436, 17 pages
http://dx.doi.org/10.1155/2011/593436
Research Article

Analysis of Compactly Supported Nonstationary Biorthogonal Wavelet Systems Based on Exponential B-Splines

1Department of Mathematical Sciences, KAIST, 373-1 Guseong-dong, Yuseong-gu, Daejeon 305-701, Republic of Korea
2Department of Mathematics, Ewha Womans University, Seoul 120-750, Republic of Korea

Received 1 June 2011; Accepted 30 September 2011

Academic Editor: Agacik Zafer

Copyright © 2011 Yeon Ju Lee and Jungho Yoon. 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

This paper is concerned with analyzing the mathematical properties, such as the regularity and stability of nonstationary biorthogonal wavelet systems based on exponential B-splines. We first discuss the biorthogonality condition of the nonstationary refinable functions, and then we show that the refinable functions based on exponential B-splines have the same regularities as the ones based on the polynomial B-splines of the corresponding orders. In the context of nonstationary wavelets, the stability of wavelet bases is not implied by the stability of a refinable function. For this reason, we prove that the suggested nonstationary wavelets form Riesz bases for the space that they generate.

1. Introduction

For the last two decades, the wavelet transforms have become very useful tools in a variety of applications such as signal and image processing and numerical computation. The construction of classical wavelets is now well understood thanks to such pioneering works as [13]. Many properties, such as symmetry (or antisymmetry), vanishing moments, regularity, and short support, are required in a practical use for application areas. In particular, polynomial splines have been a common source for wavelet construction [1, 36]. A new class of compactly supported biorthogonal wavelet systems that are constructed from pseudosplines was introduced in [7].

Exponential B-splines and polynomials have been found to be quite useful in a number of applications such as computer-aided geometric design, shape-preserving curve fitting, and signal interpolation [810]. Exponential B-splines were used as a key ingredient for the construction of wavelets [11, 12] and particularly used in wavelet construction on and [13]. In particular, in the approximation and sparse representation of acoustic signals, polynomial-based (stationary) wavelet systems have an important limitation because they do not consider the spectral features (e.g., band limited) of a given signal. However, (non-stationary) wavelet systems based on the exponential B-spline can be tuned to the specific trait of the given signal, yielding better approximations and sparser representations than classical wavelets at strictly the same computational costs. Details on exponential splines can be found in the selected references [10, 11, 1416]. Related studies on non-stationary wavelets can be found in [11, 12, 15, 1722].

One natural and convenient way to introduce wavelets is to follow the notion of multiresolution analysis. However, because the refinement masks we are interested in are non-stationary (i.e., scale dependent), we use the structure of non-stationary multiresolution analysis as introduced in [17]. Given an integer and compactly supported refinable functions , , in , we say that a sequence of subspaces forms a non-stationary multiresolution analysis (MRA) of if the following conditions are satisfied: (1) for all ,(2) is dense in ; (3)the set is a Riesz (or stable) basis for for each .

The nested embedding of the spaces implies the existence of a sequence that satisfies the non-stationary refinement equation where the sequence is usually called the refinement mask for . One should notice that the function is no longer a dilated version of in . A refinable function with the mask is called the dual refinable function of (or just the dual of for simplicity) if it satisfies Let and be a pair of MRAs generated by a pair of dual refinable functions and , , respectively. The concept of biorthogonal wavelets is to find complement spaces and of and , respectively, satisfying , and , . The corresponding biorthogonal wavelets are given by

A generalization of the biorthogonal wavelets of Cohen-Daubechies-Feuveau [1] was introduced that was based on exponential B-splines [12]. By generalizing the Strang-Fix conditions, the authors discussed the relationship between the reproduction of exponential polynomials (by or ) and the zeros of the corresponding Laurent polynomials. They also proved that for each , the proposed non-stationary refinable function generates a Riesz basis for and that the corresponding Riesz (upper and lower) bounds are independent of . However, the authors did not explicitly address the biorthogonality condition of the corresponding non-stationary refinable functions. Moreover, some fundamental questions concerning the global stability, and regularity, were left unanswered. Therefore, the primary goal of this paper is to address these issues. First, we provide a sufficient condition for the biorthogonality (1.3) of non-stationary refinable functions, and then we prove that the refinable functions based on exponential B-splines have the same regularities as the ones based on the polynomial B-splines of the corresponding orders. In the context of non-stationary wavelets, the stability of the wavelet bases is not implied by the stability of a refinable function. Therefore, we prove that the set forms a Riesz basis for the space . Furthermore, we show that the set becomes a Riesz basis for the space .

This paper is organized as follows. In Section 2, we provide basic notions of exponential B-splines. Section 3 discusses the biorthogonality condition of non-stationary refinable functions and then studies their regularities. In Section 4, we prove the (global) stability of the proposed non-stationary wavelet bases.

2. Preliminaries: Exponential B-Splines

Given a set of complex numbers , the corresponding th-order exponential B-splines can be defined as successive convolutions of the first-order B-spline with a normalization factor defined so that (see [10]), where indicates the first-order -spline, that is, , and , , is the exponential function. For simplicity, we will omit in . Obviously, the function is a compactly supported piecewise exponential polynomial. The global regularity of is (see [10, 15]). A convenient way to represent an (th-order) exponential B-spline is with the Laurent polynomial where is the normalization factor defined by We call the symbol of . Throughout this paper, is assumed to be a real or a pure imaginary number, that is, or . Since we want the mask to be symmetric with respect to its center, it is reasonable to assume that if , then . The relationship between the reproduction of exponential polynomials by and the zeros of the corresponding Laurent polynomial is discussed in [12].

It is well known that the integer translates , , are linearly independent if and only if for [10, 16]. From (2.1) and (2.2), we can easily deduce that for each , the integer translates , , are linearly independent if and only if A concept related to (but weaker than) the linear independence is the notion of the stability of . It is known (see, e.g., [23]) that the set forms a Riesz basis if and only if there exist constants such that where the bracket product for is defined by We say that the function is stable if (2.5) is satisfied. If the integer translates of are linearly independent, the stability of is immediate [24, Theorem  1.2], that is, is a Riesz basis for . Finally, the basic requirement on the set can be summarized as follows: (i) each is a real or a pure imaginary number; (ii) both and belong to , and (iii) for any , with .

3. Dual Refinable Functions

3.1. Construction of Dual Refinable Functions

Given refinable functions with the first step in the construction of a biorthogonal wavelet system is to find their dual refinable functions , (whose symbol is denoted by ). A necessary condition for and to satisfy (1.3) is where is the conjugate of the complex number . Thus, the construction of starts with the construction of a dual symbol such that (3.1) is satisfied. The algorithm to find is analogous to the classical method [1], except for the main difference that , , are scale dependent. However, in order to ensure the existence of satisfying (3.1), we need to prove that has no roots of opposite signs. For this purpose, denoting and setting and (i.e., ), we represent as follows: where Indeed, we are looking for a Laurent polynomial of the form such that (3.1) holds. Therefore, if we define , that is, then the problem of finding is reduced to constructing which satisfies the equation Let us now prove that there is no common zero of and on .

Proposition 3.1. Let the polynomials , , be given as in (3.5). Assume that for any , for any with , then and have no common roots on .

Proof. Assume that and have a common root on . This is equivalent to the existence of a number such that From (2.2), we can deduce that for some , and . It follows that , which contradicts the initial assumption.

By virtue of Proposition 3.1, the Bezout theorem guarantees the existence of a unique polynomial of degree that satisfies (3.6). One may look for a polynomial of degree higher than . However, if the corresponding dual refinable functions are to have the shortest possible support, the degree of must be constrained to be . On the other hand, it is of interest to see that by regrouping the simple fractions of into two groups, that is, then we can define the adjusted Laurent polynomials and by so that the lengths of and are very close. This allows us to construct generalized non-stationary refinable functions and . If , the resulting function becomes an exponential B-spline of order . On the other hand, the classical counterparts of and (which are obtained by setting for all ) can be written as For notational simplicity, we will write

Lemma 3.2. Let and with , then, as tends to , one has .

Proof. This is a direct consequence of [12, Lemma  2].

For the given Laurent polynomials with , there corresponds a potential candidate for the refinable function which is defined in terms of the Fourier transform as where provided that as . In fact, in (3.12) is the only candidate for the refinable function associated with such that is a dual of . Although the infinite product in (3.12) converges pointwise and is in whenever is in [12], it still needs to be ensured that the function in (3.12) is indeed a dual of . The following results address this issue. For simplicity, using and in (3.9), we introduce the notation

Lemma 3.3. Let , , be given as in (3.12) with the symbol in (3.9). Suppose that for some integer , , then which implies that . Moreover, the function can be defined as in (3.12), then this lemma also holds for .

Proof. Let and define by For the given with , set . Then, we get the identity Here, we can see that there exist and such that if and , . Also, it is obvious that for some constant independent of but dependent on . Therefore, we have with a constant depending on . Next, consider the case of and let be a sufficiently small number so that . Noting that for all , it is not difficult to see that for a sufficiently large , then we obtain that for a constant depending on . Consequently, invoking the fact that with in (3.9), we obtain for any . Here, using the same argument in [1, Proposition  4.8] (see point 3), we can get This together with (3.21) implies that for any , where is independent of . Since pointwise as , we get the relation in (3.15) with . For , applying an inductive argument based on the refinement equation , we obtain the required result. The case of can be done similarly.

Lemma 3.4. Let and , , be given as in (3.9). Suppose that for some integers , with and in (3.14), then, for any , one has for all .

Proof. Recalling the definition of in (3.16), define by Then and converge pointwise to and , respectively. Moreover, using (3.1), we can derive the relation Repeating this process yields the identity . By Lemma 3.3, , then it is immediate from the Lebesgue-dominated convergence theorem that and converge to and , respectively, in . This in turn implies that in , as . Applying Plancherel’s theorem, we arrive at the conclusion that for any .

This result proves that , , for some (in fact, for a sufficiently large ) which guarantees the condition (3.23). But the following proposition indeed proves that this duality condition holds for any , under some suitable condition on the symbols and in (3.10). In the following analysis, it is useful to use the notation with and in (3.10).

Proposition 3.5. Let and , , be given as in (3.9). Assume that for some integers , then, for any , one has for all .

Proof. Due to Lemma 3.2, we find that as , and converge uniformly on to and , respectively. Thus, we can deduce that there exists a large such that It follows from Lemma 3.4 that for any , for all . For the case , this property can be derived by using an inductive argument based on the non-stationary refinement equation. Specifically, applying (1.2), we get for any . This completes the proof.

3.2. Smoothness of Refinable Functions

For a given with and , the Hölder space (e.g., see [3]) is defined to be the space of -times continuously differentiable functions whose th derivative satisfies the Lipschitz condition In particular, the regularity of also can be analyzed by estimating the decay of around . If , , for an arbitrary small , then belongs to the space . In the following theorem, under some more stringent condition on (than (3.27)), we derive the smoothness of non-stationary refinable functions , , in (3.12). The smoothness of can be shown in a similar way.

Theorem 3.6. Assume that the Laurent polynomial in (3.10) can be rewritten as such that for some . Let , , be given as in (3.12) with the symbol in (3.9), then for some .

Proof. The Laurent polynomial in (3.9) can be written as From Lemma 3.2, we can deduce that uniformly on as . This implies that in (3.14) converges to () as . Thus, there exists a sufficiently large such that for any , which means that . Hence, recalling from (3.15) that we see that for any , with . Next, consider the case of . By applying an inductive argument based on the refinement equation we show that for any .

Remark 3.7. It is known (e.g., see [1]) that the Fourier transform of with the symbol in (3.31) has the decay rate Since converges to as tends to , we can deduce that the functions , , have the same regularity as . For the details about the (optimal) regularity of , the reader is referred to [1]. One may investigate the regularity of by using the concept of asymptotical equivalent subdivision schemes (see [15, 19]). However, the methods in [15, 19] are mainly concerned with the integer smoothness of the refinable functions.

4. Stability of Nonstationary Biorthogonal Wavelets

For each , if the condition (2.4) holds, the integer translates , , are linearly independent, and in particular, the set forms a Riesz basis for , that is, there exist constants such that In [12], it was proved that there exists some constants independent of (but dependent on ) such that However, in the context of non-stationary wavelets, the stability of does not imply the global stability of the wavelet bases , where The results in this section fill the gap. Specifically, we prove that the set is a Riesz basis for . Further, we show that the set becomes a Riesz basis for the space .

In the sequel, we will use the notation , when referring to the pair of dual refinable functions based on the th-order polynomial B-spline, and , for their corresponding biorthogonal wavelets (i.e., for all ).

Lemma 4.1. Let and be a pair of biorthogonal wavelet functions associated with and , , then (resp., ) converges to (resp., ) in , as , with the convergence rate .

Proof. With the refinement masks and of and , respectively, it is apparent from (1.4) that It has been proved in the proof of Theorem  3 in [12] that as . Thus, by Lemma 3.2, it is obvious that . Similarly, the convergence of to (as ) can be proved.

Proposition 4.2. Assume that and are a pair of biorthogonal wavelet functions associated with and , . Let and , then, for any , the sets and form Riesz bases for and , respectively. Furthermore, there exist constants independent of (but dependent on ) such that

Proof. Since and are compactly supported functions, there exist constants such that Notice that the duality condition is equivalent to , then, by using the Cauchy-Schwartz inequality, we get Putting , we obtain from (4.7) that . Next, we prove that the Riesz bounds and , , are in some interval with . For this, we see that the periodic function is of the form Here, by Lemma 4.1, converges to in as . Thus, it follows that for any , converge to as . Moreover, since is compactly supported, only a finite number of is nonzero. It yields that Therefore, we can deduce that the Riesz bounds and converge to and (resp.), as , which are the Riesz (upper and lower) bounds of . It concludes that with some constants independent of . In a similar fashion, it can be proved that with some constants independent of .

The above proposition discusses the stability of wavelet functions at each fixed level. The real problem is the global stability of the set (resp., ) with in (4.3). The following results treat this problem.

Lemma 4.3. Let and be a pair of biorthogonal wavelet functions associated with and , , then there exist two constants depending on such that for all , where is given in (4.3).

Proof. Let , and invoke that has the same support as for any . Put with , then, we observe that where the last bound is obtained by Lemma 4.1. Then, we get Putting all together, it is immediate that for some constant , where is the Riesz upper bound of . Similarly, we can prove the second relation in (4.12).

We now arrive at the central results of this section.

Theorem 4.4. Let and be a pair of biorthogonal wavelet functions associated with and , , then the sets and form Riesz bases for the spaces and , respectively, where is given in (4.3).

Proof. Let be a function in the space . By the Cauchy-Schwartz inequality and Lemma 4.3, we obtain for some constant . Putting , we have Since the upper bound is proved in Lemma 4.3, it concludes that for some constant . Further, it is obvious that , , are linearly independent. It indeed proves that the set constitutes a Riesz basis for the space (see [4, Theorem  3.20]). Similarly, we can show that forms a Riesz basis for the space .

Since and in as , it is immediate that for any , the biorthogonal projection converges to in the -norm as , then based on this observation, we get the following result.

Theorem 4.5. Let and be a pair of biorthogonal wavelet functions associated with and , , then the set forms a Riesz basis for . This result also applies to the set of dual functions .

Proof. For a given function , for notational simplicity, we define the following sequences: Similarly, let and , , be the sequences consisting of the terms and , , respectively. It has been proven in [12] that there exist constants such that Also, by Theorem 4.4, for some constant , then, by the Cauchy-Schwartz inequality and Theorem 4.4, we get Putting , we have Since the upper bound is clear from Lemma 4.3, it concludes that for some constant . It proves that the set constitutes a Riesz basis for (see [25, Theorem  6.1.1]). Similarly, we can show that forms a Riesz basis for .

Acknowledgments

The authors are grateful to the anonymous referees for their valuable suggestions on this paper. The work of Y. J. Lee was supported by Basic Science Research Program (2009-0068156) and J. Yoon was supported by Mid-Career Researcher Program (2009-0084583) and Basic Science Research Program (2010-0016257), through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology.

References

  1. A. Cohen, I. Daubechies, and J.-C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Communications on Pure and Applied Mathematics, vol. 45, no. 5, pp. 485–560, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Communications on Pure and Applied Mathematics, vol. 41, no. 7, pp. 909–996, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. I. Daubechies, Ten Lectures on Wavelets, vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1992.
  4. C. K. Chui, An Introduction to Wavelets, vol. 1 of Wavelet Analysis and Its Applications, Academic Press, Boston, Mass, USA, 1992.
  5. C. K. Chui and J.-z. Wang, “A cardinal spline approach to wavelets,” Proceedings of the American Mathematical Society, vol. 113, no. 3, pp. 785–793, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. C. K. Chui and J.-Z. Wang, “On compactly supported spline wavelets and a duality principle,” Transactions of the American Mathematical Society, vol. 330, no. 2, pp. 903–915, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. B. Dong and Z. Shen, “Construction of biorthogonal wavelets from pseudo-splines,” Journal of Approximation Theory, vol. 138, no. 2, pp. 211–231, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. N. Dyn, D. Levin, and A. Luzzatto, “Refining oscillatory signals by non-stationary subdivision schemes,” in Modern Developments in Multivariate Approximation, vol. 145 of International Series of Numerical Mathematics, pp. 125–142, Birkhäuser, Basel, Switzerland, 2003. View at Google Scholar · View at Zentralblatt MATH
  9. M. K. Jena, P. Shunmugaraj, and P. C. Das, “A subdivision algorithm for trigonometric spline curves,” Computer Aided Geometric Design, vol. 19, no. 1, pp. 71–88, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. M. Unser and T. Blu, “Cardinal exponential splines. I. Theory and filtering algorithms,” IEEE Transactions on Signal Processing, vol. 53, no. 4, pp. 1425–1438, 2005. View at Publisher · View at Google Scholar
  11. I. Khalidov and M. Unser, “From differential equations to the construction of new wavelet-like bases,” IEEE Transactions on Signal Processing, vol. 54, no. 4, pp. 1256–1267, 2006. View at Publisher · View at Google Scholar
  12. C. Vonesch, T. Blu, and M. Unser, “Generalized Daubechies wavelet families,” IEEE Transactions on Signal Processing, vol. 55, no. 9, pp. 4415–4429, 2007. View at Publisher · View at Google Scholar
  13. S. Dahlke, W. Dahmen, I. Weinreich, and E. Schmitt, “Multiresolution analysis and wavelets on 𝕊2 and 𝕊3,” Numerical Functional Analysis and Optimization, vol. 16, no. 1-2, pp. 19–41, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. W. Dahmen and C. A. Micchelli, “On multivariate E-splines,” Advances in Mathematics, vol. 76, no. 1, pp. 33–93, 1989. View at Publisher · View at Google Scholar
  15. N. Dyn and D. Levin, “Analysis of asymptotically equivalent binary subdivision schemes,” Journal of Mathematical Analysis and Applications, vol. 193, no. 2, pp. 594–621, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. A. Ron, “Linear independence of the translates of an exponential box spline,” The Rocky Mountain Journal of Mathematics, vol. 22, no. 1, pp. 331–351, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. C. de Boor, R. A. DeVore, and A. Ron, “On the construction of multivariate (pre)wavelets,” Constructive Approximation, vol. 9, no. 2-3, pp. 123–166, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. A. Cohen and N. Dyn, “Nonstationary subdivision schemes and multiresolution analysis,” SIAM Journal on Mathematical Analysis, vol. 27, no. 6, pp. 1745–1769, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. N. Dyn, D. Levin, and J. Yoon, “Analysis of univariate nonstationary subdivision schemes with application to Gaussian-based interpolatory schemes,” SIAM Journal on Mathematical Analysis, vol. 39, no. 2, pp. 470–488, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. T. N. T. Goodman and S. L. Lee, “Convergence of nonstationary cascade algorithms,” Numerische Mathematik, vol. 84, no. 1, pp. 1–33, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. B. Han and Z. Shen, “Compactly supported symmetric C wavelets with spectral approximation order,” SIAM Journal on Mathematical Analysis, vol. 40, no. 3, pp. 905–938, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. J. Prestin and E. Quak, “Trigonometric interpolation and wavelet decompositions,” Numerical Algorithms, vol. 9, no. 3-4, pp. 293–317, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. R. Q. Jia and C. A. Micchelli, “Using the refinement equations for the construction of pre-wavelets. II. Powers of two,” in Curves and Surfaces (Chamonix-Mont-Blanc, 1990), pp. 209–246, Academic Press, Boston, Mass, USA, 1991. View at Google Scholar
  24. R. Q. Jia and J. Wang, “Stability and linear independence associated with wavelet decompositions,” Proceedings of the American Mathematical Society, vol. 117, no. 4, pp. 1115–1124, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. O. Christensen, An Introduction to Frames and Riesz Bases, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, Mass, USA, 2003.