Abstract
We obtain a family of refinable functions based on generalized Bernstein polynomials to provide derived properties. The convergence of cascade algorithms associated with the new masks is proved, which guarantees the existence of refinable functions. Then, we analyze the symmetry, regularity, and approximation order of the refinable functions, which are of importance. Tight and sibling frames are constructed and interorthogonality of sibling frames is demonstrated. Finally, we give numerical examples to explicitly illustrate the construction of the proposed approach.
1. Introduction
Because it is highly desirable to construct wavelets within a class of analytically representable functions, compactly supported sibling frames with interorthogonality attract a considerable amount of attention, recently.
In 1997, Ron and Shen completed the structure of the affine system, which can be factored during a multiresolution analysis construction. This leads to a characterization of all tight frames that can be constructed by the methods in [1]. In 2000, compactly supported tight frames that correspond to refinable functions were studied and a constructive proof was given by Chui and He [2]. In [3], Han gave his investigation of symmetric tight framelet filter banks with a minimum number of generators and systematically studied them with three high-pass filters which are derived from the oblique extension principle. In 2002, compactly supported tight and sibling frames, with symmetry (or antisymmetry), minimum support, shift-invariance, and interorthogonality, were constructed in [4]. In 2003, Daubechies et al. discussed wavelet frames constructed via multiresolution analysis, with emphasis on tight wavelet frames. More importantly, they established general principles and specific algorithms for constructing framelets and tight framelets in [5]. In 2005, Averbuch et al. [6] obtained tight and sibling frames originated from discrete splines, in which, all the filters are linear phase and generate symmetric scaling functions with analysis and synthesis pairs of framelets. Next, in [7], symmetric wavelets dyadic sibling and dual frames, where each of the frames consists of three generators obtained using spectral factorization, were given. In 2007, a new type of pseudo-splines was introduced to construct symmetric or antisymmetric tight framelets with desired approximation orders by Dong and Shen [8]. And they provided various constructions of wavelets and framelets. In 2013, Shen and Xu [9] give -Spline framelets derived from the unitary extension principle, which led to the result that the wavelet system is generated by finitely many consecutive derivatives. More tight frames have been studied in [10–20], so far.
This paper is concerned with the construction of compactly supported tight and sibling frames based on generalized Bernstein polynomials [21], defined aswhere . We complete the convergence of cascade algorithms associated with the new masks. Furthermore, the symmetry, regularity, and approximation orders of corresponding refinable functions are analyzed. At last, we implement interorthogonality of sibling frames.
The remainder of this paper is organized as follows. In Section 2, some notations about refinement marks are collected and some technical lemmata are given to use in other sections. We will elaborate on convergence of cascade algorithms based on the masks, which guarantees the existence of refinable functions in Section 3. Section 4 analyzes the symmetry and gives a symmetry proof. In Section 5, regularity and approximation orders are focused on study; at the same time, we obtain the lower bound of the regularity exponents of refinable functions by estimating the decay rates of their Fourier transform. At last, we construct tight and sibling frames and obtain interorthogonality of sibling frames in Section 6.
2. Preliminaries
For the convenience of the readers, we review some definitions and properties about refinement marks in this section.
New marks based on generalized Bernstein polynomials (1), with order , for given nonnegative integers , , and , are defined as follows:
For notational simplicity, we will introduce the following two definitions:
By , we denote all the functions satisfying and the set of all sequences defined on such that
In the following, we will give a compactly supported real-valued refinable function with finite mask and real mask coefficients; that is, satisfies a two-scale relation:for some real numbers . Assume that the corresponding two-scale Laurent polynomialssatisfyfor some , with a Laurent polynomial that satisfies .
The Fourier transform of is And, satisfies
With the above, the refinement equation (6) can be written in terms of its Fourier transform aswhere , . We call the refinement mask for convenience, too.
By the iteration of (11), the corresponding refinable function can be written in terms of its Fourier transform as
In the following, we will adopt some of the notations from [2, 4, 22]. The transition operator for -periodic functions and can be defined as
For , a quantity is defined by
The notation is defined by
For convenience, assume that
The family is interorthogonal if , , where span .
The modulus of continuity of a function defined on an open interval will be denoted, as usual, by A function defined on the real line is called a piecewise Lip function, , if there exist finitely many values , such that where , .
Two finite families, , are defined by scaling relationswhere , are Laurent polynomials that have real coefficients and vanish at . In other words, where . Hence, the functions and have compact support and at least one vanishing moment.
A function belongs to the Hölder class with , if is a -periodic continuous function such that is times continuously differentiable and there exists a positive number satisfying for all , where is the largest integer such that .
We use for approximation of . And a function satisfies the Strang-Fix condition of order if
Under certain conditions on (e.g., if it is compactly supported and ), the Strang-Fix condition is equal to the requirement that has a zero of order at each of the points in In [5], if satisfies the Strang-Fix condition of order and the corresponding mark satisfies that at the origin, then the approximation order is .
We will provide some lemmas which are necessary for the following theorem. The following lemmas are about the relations of the quantities associated with masks and a condition of the convergence of cascade algorithms.
Lemma 1 (see [22, Theorem ]). Let be a -periodic measurable function such that with and . If a.e. for some such that , then
Lemma 2 (see [22, Theorem ]). Let and be -periodic measurable functions such that for almost every . Then
Lemma 3 (see [22, Theorem ]). Let with and . If , then the cascade algorithm associated with the mask converges in the space .
For regularity, our primary goal is to obtain the lower bound of its exponents of refinable functions by estimating the decay rates of their Fourier transform. The relation is expressed by for any small enough ; see [23]. Consequently, . Next, we will give an estimate of the decay rates of the Fourier transform of refinable functions with the mask . By [23, 24], for any stable, compactly supported refinable functions in with , the refinement mask must satisfy and . Thus, can be factorized as where is the maximal multiplicity of the zeros of at and is a trigonometric polynomial with . Therefore, one obtains which shows the decay of can be characterized by as stated in the following lemma.
Lemma 4. Let be the refinement mask of the refinable function of the form Ifthen with , and this decay is optimal.
The following lemmas are useful for obtaining the important tight and sibling frames.
Lemma 5 (see [4, Theorem ]). For any compactly supported refinable function that satisfies (8)–(16), there exist compactly supported sibling frames , with the property that all of the four functions have vanishing moments, where is the order of the root of the two-scale Laurent polynomial P. Furthermore, if is symmetric, then all of the four functions can be chosen to be symmetric for even and antisymmetric for odd .
Lemma 6 (see [4, Theorem ]). For any compactly supported refinable function that satisfies (8)–(16), there exists a pair of sibling frames and such that all of the four functions have compact support and the maximum number of vanishing moments and that is interorthogonal.
Lemma 7 (see [4, Theorem ]). Let , be a pair of compactly supported sibling frames associated with a VMR function . If is Laurent polynomial, then the function with two-scale symbol , where , defines a tight frame of which is associated with the same VMR function .
3. Convergence of Cascade Algorithms Based on the Masks
In this section, demonstration of the convergence of cascade algorithms in the space is given. To complete it, a useful condition of proving the convergence of cascade algorithms is described as follows.
Lemma 8. For two positive integers , , ifthen
Proof. For , it is obvious that We claim thatSince , for , it holds thatThere are two cases to consider.
Case 1. Suppose that . By (32) and (36), it is easy to see that ThenThis implies condition (35).
Case 2. Suppose that . In the same way, we get for . Then (38) holds. This concludes claim (35). By using (32), one gets ThenThus, Similarly, one hasFor any , notice thatand , which is a continuous function on and is differentiable on , has the maximum value at . The reason is as follows: the equation has three zeros, at . Since , is the minimum of on . Thus, is the maximum of on . Therefore, applying (35), (41), (43), (44), and we get inequality (33).
Theorem 9. For two positive integers , satisfying , if we let be mask (2), then the cascade algorithm associated with the mask converges in the space .
Proof. For , one has Applying and Lemma 8, we obtainBringing Lemmas 1 and 2 together yields Thus, by Lemma 3, the cascade algorithm associated with the mark converges in the space .
4. Symmetry
Symmetric coefficients of the mark are of great significance in image processing. The following lemma is helpful for the demonstration of symmetry.
Lemma 10. For , , , , we derive wherewhere
Proof. Set ; then andFixing , by using (54), we obtainSince coefficient determinant of (55) applying Cramer’s Rule yields where
In the following, we will give a symmetry proof.
Theorem 11. For two positive integers , , satisfying , let be mask (2); then the coefficients of the mask are symmetric.
Proof. Let and thenSince , , we set and obtain Let ; by using Lemma 10, one can obtain that where is (51) in Lemma 10. Let and ; then We consider two cases. Suppose that is an even number. Applyingyields where ,Thus, where ,Suppose, on the other hand, that is an odd number. It holds where ,Therefore, where ,This establishes the proof.
5. Regularity and Approximation Orders
This section is devoted to analysis of the regularity and approximation orders of refinable functions with the mask defined by (2) in the following theorem.
Theorem 12. For two positive integers , , satisfying , let be refinable functions with the mask . Then where , and the decay rate is optimal. As a result, , where , for any small enough .
Proof. It is obvious that , . We claim thatIndeed, is a continuous function on and is differentiable on . The maximum value of on can be derived as follows: we find that is the only zero of equation . Obviously, is the minimum of on . Consequently, . Here, is an even function. Hence, (74) holds. Next we showHere, is a continuous function on and is differentiable on . Since equation has no zeros on , we are required to compare with . Because of we get . Thus, (75) holds. By Lemma 4, satisfies where and the decay rate is optimal. As a result, , where , for any small enough .
Theorem 13. For two positive integers , , satisfying , let be refinable functions with the mask . Then provides the approximation orders .
Proof. In fact, the approximation orders of and are independent on . For convenience, we set . Following [8], let where . Since is equal to when , and has zero of order , we conclude thatFor then provides the approximation orders .
6. Tight and Sibling Frames
In this section, tight and sibling frames are constructed in the following theorem. At the same time, the interorthogonality of sibling frames is implemented.
Theorem 14. For two positive integers , satisfying , let be mask (2), then there exist compactly supported sibling frames , defined in (19), with the property that all of the four functions have 2 vanishing moments. Furthermore, all of the four functions are symmetric.
Proof. Let : We getHere, .
Obviously,In [2], another way of looking at this is that the decay condition for some and , automatically implies that , for .
Applying Theorem 12 yields Then, it is obvious thatHence, by Lemma 5 and (83)–(87), there exist compactly supported sibling frames , defined in (19).
Theorem 15. For two positive integers , satisfying , let be mask (2); then there exists a pair of sibling frames and , with the property that all of the four functions have compact support and the maximum number of vanishing moments. And, is interorthogonal.
Proof. We claim that condition (8)–(16) holds. Indeed, the refinable functions with the masks , which satisfy (8)–(16) as shown in Theorem 14, have been proved. Applying Lemma 6 yields the conclusion.
Theorem 16. For two positive integers , , satisfying , let be mask (2): , where is Laurent polynomial. Then defines a tight frame of which is associated with the same VMR function .
Proof. Since is mask (2), by Theorem 14, there exist compactly supported sibling frames , . Then let , associate with a VMR function , where is Laurent polynomial.
Applying Lemma 7, then construct a tight frame of which is associated with the same VMR function .
7. Numerical Example
Consider Theorem 16 with the condition as follows.
Example 1 (see Figure 1). Take The corresponding refinable function is order . Let , and get in Theorem 16.
(a)
(b)
Example 2 (see Figure 2). Take The corresponding refinable function is order . Let , and get in Theorem 16.
(a)
(b)
In Figures 1 and 2, we give the examples of tight and sibling frames with one generator; at the same time, their convergence of cascade algorithms, symmetry, and approximation orders is analyzed.
8. Conclusions
In this paper, we study new marks (2) based on generalized Bernstein polynomials (1): with two positive integers , , satisfying , to provide derived properties. The convergence of cascade algorithms in Theorem 9 is obtained, which guarantees the existence of refinable functions. In Theorem 11, we analyze the symmetry of the refinable functions, which is of importance. The regularity and approximation order of the new refinable functions are given; at the same time, the lower bound of the regularity exponents of refinable functions is showed by estimating the decay rates of their Fourier transform. Finally, we construct tight and sibling frames and demonstrate interorthogonality of sibling frames in Section 6. And, numerical examples are given to illustrate the construction of the proposed approach.
Competing Interests
The authors declare that they have no competing interests.
Authors’ Contributions
All authors contributed equally in this paper. They read and approved the final paper.
Acknowledgments
This research is supported by the National Natural Science Foundation of China (no. 61271010) and Beijing Natural Science Foundation (no. 4152029).