Research Article | Open Access
Polynomial Reproduction of Vector Subdivision Schemes
We discuss the polynomial reproduction of vector subdivision schemes with general integer dilation . We first present a simple algebraic condition for polynomial reproduction of such schemes with standard subdivision symbol. We then extend it to general subdivision symbol satisfying certain order of sum rules. We also illustrate our results with several examples. Our results show that such kind of scheme can produce exactly the same scalar polynomial from which the data is sampled by convolving with a finite nonzero sequence of vectors.
Subdivision schemes are efficient algorithmic methods in approximation theory, computer aid geometric design, and wavelet construction; see [1, 2] and references therein. The subdivision scheme is called vector subdivision scheme if the corresponding subdivision symbol is a matrix of trigonometric polynomial. Note that the vector subdivision scheme plays an important role in multiwavelet and multichannel wavelets analysis; see [3–5] and references therein. In this paper we would like to derive simple algebraic conditions on the subdivision symbol that allow us to determine the degree of its polynomial reproduction, which is different from polynomial generalization.
The polynomial generalization is the capability of vector subdivision scheme to generate the full space of scalar polynomials with desired degree. This property is equivalent to approximation order (defined in -norm), accuracy, and sum rulers; see [6, 7]. The so-called polynomial reproduction is the capability of vector subdivision scheme to produce exactly the same scalar polynomial from which the data is sampled by convolving with a special sequence of vectors , where is some finite index set containing , denotes convolution operator, and denotes transpose operator.
There are many valuable works that were done when scalar subdivision scheme was considered [8–11]. Hormann and Sabin  presented some algebraic conditions which can be used to determine the degree of polynomial reproduction for a family of schemes. Polynomial reproduction was studied by Dyn et al.  for arbitrary prime and dual binary schemes and extended to univariate subdivision scheme of any arity by Conti and Hormann . Further, Charina and Conti  yielded simple algebraic conditions on the subdivision symbol for the multivariate setting of scalar subdivision with dilation matrix , ; Charina and Romani  extended to the general expanding dilation matrix case. Motivated by all these works, we discuss the property of polynomial reproduction for vector subdivision schemes.
The remainder of this paper is organized as follows. Section 2 sets the notations concerning vector subdivision scheme with general integer dilation and stresses the connection of convergence between the vector subdivision scheme and traditional cascade algorithm. In Section 3, we provide algebraic tools for determining the degree of polynomial reproduction of the vector subdivision scheme with standard subdivision symbol; basing on two scale similarity transform (TST), we also establish the relationship of vector subdivision schemes between general subdivision symbol and the standard subdivision symbol. In Section 4, several examples are provided to illustrate our results.
2. Background and Notation
A vector subdivision scheme with integer dilation is given by a finitely supported matrix sequence (so-called subdivision mask), a constant column vector , and an initial function . The constant vector and the initial function are determined by and will be given by (39) or (55) in the next section. The symbols of and (denoted by and , resp.) are given by the Laurent polynomials respectively.
In this paper, we use two vector spaces of Laurent polynomial, and . We say , if with some and , where is some finite -vector sequence and is the identity matrix. Similarly, we say belongs to if .
For a data sequence , the vector subdivision operator acting on it is defined by Actually the vector subdivision scheme is a recursive algorithm based on the vector subdivision operator and a convolution operator with the initial data and the function where is the characteristic function of interval and is selected such that and .
Lemma 1. If , then there exist linearly independent vectors such that
Proof. Suppose that . It is not difficult to find that there exist linearly independent constant vectors satisfying Let Then, by calculation the conclusion follows directly.
In order to define a sequence of continuous functions, we also need parameterization. As in  for the scalar case, we chose the associated parameter values: We call the sequence of parameter values associated with the vector subdivision scheme. Let us define continuous functions by linearly interpolating the data to the parameter values ,
Definition 2. If the sequence of continuous functions converges (in uniform norm), then we define the limit function as
We say that is the limit function of the vector subdivision scheme acting on the initial data . If and , one may find that the vector subdivision scheme is the same as the scalar one in . The limit functions , , are called basic limit functions if the initial data is selected by where denotes th column of the identity matrix. It is easy to check that is compactly supported and satisfies the refinement equation
In this paper, we consider vector subdivision scheme that is convergent and nonsingular, so that if and only if . Define the linear operator (which is called cascade operator) as follows: By induction on , it is easy to verify that where is defined as (7),
Proposition 3. A vector subdivision scheme is convergent and nonsingular if and only if (15) has compactly supported solution and the integer translations of are linearly independent.
Proof. Firstly, we claim that the convergence of vector subdivision scheme is the same as that of the cascade algorithm. Actually, by induction, for the initial data , we have and Then, our declaration follows by a standard method in analysis. Note that the convergence of cascade algorithm with finite supported mask is equivalent to the existence of compactly supported solution of (15) (see ). We can obtain the desired result according to the proof of [11, Proposition 1.3].
Following [10, 11], we give the definitions of polynomial generation, polynomial reproduction, and stepwise polynomial reproduction with respect to vector subdivision scheme. We denote by the space of polynomials up to degree . Suppose , for , and let be the space of -vector sequence indexed by .
Definition 4 (polynomial generation). A convergent stationary vector subdivision scheme generates polynomials up to degree (-generating), if, for any polynomial , there exists some initial data such that .
Definition 5 (polynomial reproduction). A convergent stationary vector subdivision scheme with parameter values is reproducing polynomials up to degree (-reproducing), if, for the initial data , with any polynomial , the limit of the vector subdivision scheme satisfies .
Definition 6 (stepwise polynomial reproduction). A convergent stationary vector subdivision scheme with parameter values is stepwise reproducing polynomials up to degree (stepwise -reproducing), if, for the data , with any polynomial ,
Noting basic limit function defined by (14), it is easy to show that Thus the equivalence of polynomial generation and accuracy (approximation order, sum rulers) is clear; see [6, 7]. The following proposition shows that for a nonsingular convergent vector subdivision scheme the concepts of polynomial reproduction and stepwise polynomial reproduction are equivalent.
Proposition 7. A nonsingular convergent vector subdivision scheme is stepwise reproducing if and only if it is reproducing.
Proof. Since that , then by discrete Fourier transform we have . Thus under the stepwise polynomial reproduction, from (6) we can easily get, for , As a result, implies . The rest of necessary part one may refer to [11, Proposition 1.7]. For the sufficient part, noting (6) and (7), we have Consider . Taking and using the same methods of [11, Proposition 2.1], one obtains from (6) that Noting that , we deduce from that where . Similarly, the equation implies Following Lemma 1, we select linearly independent vectors as (10) satisfying . Then, where . Since is an invertible matrix, then is equivalent to . The conclusion follows directly.
3. Algebraic Condition for Polynomial Reproduction
In this section we firstly consider the nonsingular subdivision scheme with standard subdivision symbol in (33) and provide algebraic conditions on for checking the polynomial reproduction. With the two-scale similarity transform (TST) in the hand, we further discuss the properties of polynomial reproduction with respect to the general subdivision symbol satisfying certain order of sum rules. For TST, see [6, 13, 15, 16] and references therein.
We denote the subsymbols of a vector subdivision symbol by and remark that the th derivative of a subsymbol is where is the polynomial We further assume that where , , , and are , , , and matrices of Laurent polynomials, respectively. The subsymbols of , () are defined by , (), . We define standard subvision symbol with sum rules of order by where and , are , matrices, respectively. Similar to [10, Lemma 2.1] or [11, Proposition 2.1], we have the following proposition.
Proposition 8. The following are equivalent.(i)The vector subdivision symbol takes the form of (33).(ii)The entries of satisfy with , where and .(iii)The subsymbols of the entries of satisfy (iv)The coefficients of the entries of satisfy
Similar to [10, Lemma 4.2] or [11, Proposition 2.5], we also have the following proposition. It allows us to express the polynomial reproduction of in terms of the properties of its symbol, where takes the form of (33).
Proposition 9. Let , , and as in (31). A vector subdivision symbol satisfies if and only if
3.1. Standard Subdivision Symbol
Suppose that subdivision symbol has the standard form (33) in this subsection. Then the constant vector and initial function in can be chosen by respectively, where satisfies . We provide a simple algebraic condition for determining , which appears in (11) and guarantees the reproduction of linear polynomial. In Theorem 11, we then provide algebraic conditions on for checking the reproduction of polynomials of higher degree.
Proposition 10. Let , be defined by (39) and let take the form of (33) with . Let be a nonsingular vector subdivision scheme. Then, reproduces linear polynomials if and only if its parameter values are given by (11) with
Proof. We follow the lines of Theorem 3.1 of  or Proposition 2.3 of . According to Proposition 7, polynomial reproduction is equivalent to stepwise polynomial reproduction. Hence it is sufficient to show that implies , for any . Note that any convergent vector subdivision scheme reproduces constants; we only consider the monomial . Let and . For any and , we have
As a result, the vector subdivision scheme reproduces linear polynomials stepwisely if and only if Then, (40) follows directly.
Proof. The proof is induction. In the case the conclusion follows by Proposition 8. By Proposition 7, it suffices to prove the results for the step polynomial reproduction.
“:” Let and . Suppose that , with . We show that satisfies . Actually by induction and stepwise polynomial reproducing, we have By Proposition 9, (43) is equivalent to Then,
“:” Assume that the vector subdivision scheme is reproducing. Let and , with . Using the same method as above, we have On the other hand, Combining the above two equations, we deduce By induction, we obtain As a result, by Proposition 9 the conclusion (43) follows.
3.2. General Subdivision Symbol
We devote this section to discussing the polynomial reproduction of vector subdivision scheme with general subdivision symbol. Suppose that subdivision symbol has sum rules of order (see [6, 7]), that is, if there exists a row vector of trigonometric polynomials such that and
In [16, Theorem 2.2], they showed that subdivision symbol satisfies the sum rules of order , if and only if there exists an invertible matrix such that with taking the form (33), where is a constant invertible matrix. It is evident that with .
Let the constant vector and initial function in be defined by (39). Then we choose and in by with . It is easy to see that , , and .
Theorem 12. The vector subdivision scheme is reproducing, if and only if is reproducing.
Proof. By Proposition 7, we only need to consider the equivalence under the condition of stepwise reproducing. We just show the sufficient part (the necessary part can be proved similarly).
Suppose and , where and parameter values are defined by (11). By the definition of stepwise polynomial reproduction, we have Since , then we get , where are the coefficients of with and are defined similarly. For the new scheme , we have Then Here, necessary variable substitution is used. Following the stepwise polynomial reproduction of , we know that Thus the proof is completed.
Example 1 (see [17, Example 4.2], GHM refinable mask). The vector subdivision scheme is based on the following mask symbol,
It is not difficult to find that the subdivision mask satisfies the sum rules of order . The corresponding vector in (52) is selected by We construct a trigonometric matrix by Then, takes the form of (33), with Using Proposition 10 we get that, for , the scheme reproduces linear polynomials. Since , the scheme cannot reproduce polynomials of degree . For their graphs, see Figure 1.
Example 2 (see [17, Example 5]). The vector subdivision scheme is based on the mask symbol with
The subdivision mask satisfies the sum rules of order ; see . The corresponding vector in (52) is selected by If is chosen by then takes the form of (33), with . Using Proposition 10 we get that, for , the scheme reproduces linear polynomials. Since , the scheme cannot reproduce polynomials of degree . For their graphs, see Figure 2.
Example 3 (see [16, Example 5.2]). The vector subdivision scheme is based on the following mask symbol:
The subdivision mask satisfies the sum rules of order ; see  or . If is chosen by then takes the form of (33), with Using Proposition 10 we get that, for , the scheme reproduces linear polynomials. Since , the scheme also can reproduce polynomials up to degree . For their graphs, see Figure 3.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (Grant no. 11071152) and the Natural Science Foundation of Guangdong Province (Grant nos. 10151503101000025 and S2011010004511).
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