Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials

Abstract: In this paper, we propose a family of non-stationary combined ternary m 2 3 ( ) + -point subdivision schemes, which possesses the property of generating/reproducing high-order exponential polynomials. This scheme is obtained by adding variable parameters on the generalized ternary subdivision scheme of order 4. For such a scheme, we investigate its support and exponential polynomial generation/reproduction and get that it can generate/reproduce certain exponential polynomials with suitable choices of the parameters and reach m 2 3 + approximation order. Moreover, we discuss its smoothness and show that it can produce C m 2 2 + limit curves. Several numerical examples are given to show the performance of the schemes.


Introduction
Subdivision schemes are an efficient tool to design smooth curves and surfaces from a given initial polyline/ polyhedral mesh. Over the past two decades, subdivision schemes have shown their usefulness in several application contexts ranging from computer aided geometric design and signal/image processing to computer graphics and animation. Recently, subdivision schemes have become of interest also in biomedical imaging applications (see, e.g., [1][2][3]) and isogeometric analysis (IgA), a modern computational approach that integrates finite element analysis into conventional CAD systems (see, e.g., [4][5][6][7]). Generally, given a set of initial values on a coarse grid, a subdivision scheme is a set of rules that recursively define sets of values on finer grids. If the rules are the same at all refinement levels, then the scheme is stationary, otherwise it is non-stationary.
One of the important capabilities of non-stationary schemes (which stationary schemes do not have) is the reproduction of exponential polynomials. Such schemes may be useful for the processing of families of oscillatory signals that are well approximated by combinations of exponential polynomials, such as speech signals, and narrowband signals in general. Hence, there have been continuous works on non-stationary subdivision schemes generating exponential polynomials. Romani [8] converted three exponential B-spline schemes into interpolatory schemes without changing the generation property. Conti et al. [9] transformed the non-stationary approximating schemes into interpolatory ones with the same generation property.
For other references on this method, see, e.g., [10][11][12][13][14][15]. All of the aforementioned works convert a known approximating subdivision scheme into a new interpolatory subdivision scheme by a suitable polynomial correction of the symbol of the approximating one. Another way to construct interpolatory schemes is to derive them from approximating ones using the push-back operation presented in [16]. With this method, Lin et al. [17] derived interpolatory surface subdivision from the approximating subdivision. Zhang and Wang [18] gave a semi-stationary subdivision scheme by this operation. Novara and Romani [19] used the push-back operation and constructed a combined ternary 4-point scheme which can unify quite a number of existing approximating and interpolatory schemes. However, previous works are restricted to the stationary schemes which can only generate/reproduce algebraic polynomials. Zheng and Zhang [20] first generalized the push-back operation to the non-stationary case and constructed a non-stationary combined subdivision scheme. As ternary subdivision schemes compare favorably with their binary analogues because of generating limit functions with the same (or higher) smoothness but smaller support (see, e.g., [21]), Zhang et al. [22] generalized the push-back operation to the non-stationary ternary case and presented a non-stationary combined ternary 5-point subdivision scheme. In this paper, to generate and reproduce more general exponential polynomials, we propose a family of non-stationary combined ternary m 2 3 ( ) + -point subdivision schemes based on the generalized ternary scheme of order 4 proposed in [22]. For such a scheme, we investigate its properties, including the support, exponential polynomial generation/reproduction, approximation order and smoothness. Some examples are given to illustrate the feature of the proposed scheme.
The rest of the paper is organized as follows: Section 2 recalls some basic knowledge about subdivision schemes. Section 3 is devoted to the construction of the new family of non-stationary combined ternary m 2 3 ( ) + -point subdivision schemes, which reproduces high-order exponential polynomials. The support, exponential polynomial generation/reproduction, approximation order and smoothness are investigated in Section 4. Section 5 concludes the paper with a short summary and further research work.

Background
In this section, we review a few basic definitions and results about ternary subdivision scheme which form the basis of the rest of this paper. This paper is mainly concerned with non-stationary ternary subdivision schemes formalized as follows. Given a sequence of initial control points P P i : , the control points P j P : where S a k is the k-level subdivision operator mapping 0 ( ) ℓ to 0 ( ) ℓ , and 0 ( ) ℓ denotes the linear space of real sequences with finite support. The set of coefficients a a i : in the aforementioned equations is termed as the mask of the subdivision scheme at level k and the Laurent polynomial a z a z, where D j denotes the jth order differential operator. In the following, we recall some knowledge about the generation/reproduction of exponential polynomials.
and denote by θ τ , )} = … the set of zeros with multiplicity, satisfying  In this section, our goal is to construct a family of non-stationary combined ternary m 2 3 ( ) + -point subdivision schemes reproducing exponential polynomials, on the basis of the generalized ternary subdivision scheme of order 4 in [22].
For the reader's convenience, refinement rules of the generalized ternary subdivision scheme of order 4 are recalled here: Construction of nonstationary combined ternary schemes  911 From Proposition 2 of [25], we know v k 1 + and v k indeed satisfy the following iteration We construct the non-stationary combined ternary ( m − + , and the corresponding k-level symbol can be written as a z a z a z a z , Suppose the refinement rules in (2) do not depend on the level k, i.e., ξ ξ α α ,  . Note that, when m 1 = , it turns into the non-stationary combined ternary 5-point subdivision scheme in [22]. In particular, if α α α ξ , , , where ω k ∈ is the free parameter, it actually becomes the non-stationary ternary 5-point interpolatory C 2 subdivision scheme in [26].
4 Properties of the non-stationary combined ternary ( + ) m 2 3 -point subdivision scheme In this section, we discuss the properties of the proposed non-stationary combined ternary subdivision scheme (2), including the support, exponential polynomial generation/reproduction, approximation order and smoothness.

Support
The support of a subdivision scheme represents how far one vertex affects its neighboring points whose size directly influences local support property of the subdivision curve. In this subsection, we study the support of the proposed scheme (2).
, n 0 ≥ be the family of basic limit functions generated by the non-stationary combined ternary subdivision scheme Then the scheme has support width m 3 4 + , i.e., the basic limit functions vanish outside the interval , m m 3 4 2 Proof. Assume the supports of the k-level masks associated with a non-stationary subdivision scheme are l k r k , [ ( ) ( )], k ∈ . In fact, the supports of the limit basic functions ϕ n are proved to be included in L R , (see, e.g., [22,27]). From the k-level subsymbols of the scheme (2) in (4), we have l k m rk Hence, the supports of the limit functions are , The basic limit functions of the approximating scheme (6) are shown in Figure 1 with different parameters v 0.01 0 = (solid line), 1 (dashed line), 100 (dotted line), respectively.

Exponential polynomial generation/reproduction property
The goal of this subsection is to study the exponential polynomial generation/reproduction property of the scheme (2). We start by defining the exponential polynomial spaces EP x e r τ θ τ then, fixed t The exponential polynomial spaces EP ΓΛ t and EP ΓΛ t ′ are actually m 2 3 + and m 2 4 + dimensional spaces. The following theorem shows that the proposed scheme can reproduce high-order exponential polynomials with suitable choices of parameters ξ α α , , , defined by (7), under suitable choices of the parameters ξ α α , , , Moreover, the reproduction of exponential polynomial e tx ± implies that the scheme (2) reduces to an interpolatory one.
Proof. In view of Theorem 2.6, the scheme . For the second linear system in (9), we make a substitution as follows: Then, the second linear system (9) can be written as where Thus, by Gaussian elimination, it can be seen that the rank of A k and A g   then it is not difficult to observe that equation (12) is equivalent to , , , , , 1, 0, ,0 , 1, 0, ,0 , 1, 0, ,0 , ,1, 0, ,0 , To conclude, we show that the scheme (2) turns into an interpolatory scheme, if the scheme reproduces e tx ± . In view of Theorem 2.6, it suffices to show that a e a ε e 3, 0, , which implies that the scheme (2) reduces to an interpolatory one. □ As an example, we take m 1 = in (2) and obtain the non-stationary combined ternary 5-point subdivision scheme in [22] P P α P α P α P P P ξ P The scheme (14) can reproduce EP x x e 1, , , tx As a consequence, the conic sections such as circles, parabolas and hyperbolas can be exactly reproduced by the scheme (14), but not the curves like the cardioid. To reproduce more exponential polynomials, we take m 2 = in (2). The scheme (2) becomes the non-stationary combined ternary 7-point scheme P P α P α P α P α P α P P P ξ P P P α P α P α P α P α P the scheme (16) reproduces the exponential polynomial space EP x x e e 1, , , , which contains the exponential polynomial space EP ΓΛ 1 t . Similarly, with a suitable choice of α α α α α ξ , , , , , , the scheme (16) can reproduce the exponential polynomial space EP x x e e 1, , , ,  Figure 2 shows the limit curves generated by the subdivision scheme (16) with parameters in (17) and Construction of nonstationary combined ternary schemes  919 limacon. Figure 4 displays the reproduction of the exponential polynomial space EP ΓΛ 2,2 t by the subdivision scheme (16) with parameters in (18).

Approximation order
This subsection is devoted to the analysis of approximation order of the proposed scheme (2). For this purpose, first we recall the following definitions.  The study of the approximation order for non-stationary binary schemes was carried out in [30], and we extend it here to the case of non-stationary ternary case. Theorem 4.5 can appear trivial after reading the proof of Theorem 21 in [30].

Smoothness
In this subsection, we discuss the smoothness of the non-stationary ternary scheme (2) and get that the scheme (2) reproducing the exponential space EP ΓΛt can reach C r convergence where r m 2 2 ≤ + , when the stationary counterpart (5) of the scheme (2) is C r convergent. Before investigating the smoothness of the scheme (2), we recall the following results. , then the scheme S is C n convergent, where In the non-stationary case, approximate sum rules of order N and asymptotical similarity to a stationary C N 1 − subdivision scheme are sufficient conditions for C N 1 − convergence of non-stationary schemes (see [31]). By Corollary 1 of [31], we have the following result. Theorem 4.9. Assume that the non-stationary ternary subdivision scheme S k a k { } ∈ is said to satisfy approximate sum rules of order N , N ∈ and is asymptotic similar to a C N 1 − convergent stationary scheme S a . Then, the non-stationary ternary scheme S k In the following corollary, we study the approximate sum rules of the scheme (2). Corollary 4.10. If the non-stationary ternary subdivision scheme (2) reproduces the exponential space EP ΓΛ t , then it satisfies approximate sum rules of order m 2 3 + .
The proof of Corollary 4.10 will be given in Appendix, which is in analogue to the proof of Theorem 10 in [30]. And as a direct consequence of Theorem 4.9 and Corollary 4.10, we have the following result.
Theorem 4.11. For the non-stationary ternary subdivision scheme (2) reproducing the exponential space EP ΓΛ t , it is C r convergent with r m 2 2 ≤ + , if the asymptotical similar counterpart (5) of the scheme (2) is C r convergent.

Conclusion
In this paper, we construct a family of non-stationary combined ternary m + convergent for a suitable choice of the parameters. In the future, we may focus on the analysis of other property of the scheme, such as the shapepreserving property, and the construction of non-stationary combined ternary even-point schemes which have high-order exponential polynomial generation/reproduction.