Nonstationary subdivision schemes originated from uniform trigonometric Bspline
Abstract
The paper proposes, an algorithm to produce novel point (for any integer ) binary nonstationary subdivision scheme. It has been developed using uniform trigonometric Bspline basis functions and smoothness is being analyzed using the theory of asymptotically equivalence. The results show that the most of wellknown binary approximating schemes can be considered as the nonstationary counterpart of the proposed algorithm.
Furthermore, the schemes developed by the proposed algorithm has the ability to reproduce or regenerate the conic sections, trigonometric polynomials and trigonometric splines as well. Some examples are considered, by choosing an appropriate tension parameter , to show the usefulness.
1 Introduction
Subdivision scheme is one of the most important and significant modelling tool to create smooth curves from initial control polygon by subdividing them according to some refining rules, recursively. These refining rules take the initial polygon to produce a sequence of finer polygons converging to a smooth limiting curve.
In the field of nonstationary subdivision schemes, Beccari et al. [1] presented a 4point binary nonstationary interpolating subdivision scheme, using tension parameter, that was capable of producing certain families of conics and cubic polynomials. They also developed a 4point ternary interpolating nonstationary subdivision scheme in the same year that generate continuous limit curves showing considerable variation of shapes with a tension parameter [2]. A new family of 6point interpolatory nonstationary subdivision scheme was introduced by Conti and Romani [4]. It was presented using cubic exponential Bspline symbol generating functions that can reproduce conic sections. Conti and Romani [5] discussed algebraic conditions on nonstationary subdivision symbols for exponential polynomial reproduction.
Since, nonstationary schemes have proven to be efficient iterative algorithms to construct special classes of curves. One of the important capability is the reproduction or regeneration of trigonometric polynomials, trigonometric splines and conic sections, in particular circles, ellipses etc. So, in this article, an algorithm has been introduced to produce point binary approximating nonstationary schemes, for any integer , using the uniform trigonometric Bspline basis function of order . The proposed algorithm can be considered as the nonstationary counterpart of the well known binary approximating schemes introduced by Chakin [3] and Siddiqi with his different coauthors [9, 13, 14, 15, 16], after setting different values of in proposed algorithm (for details see table 5.1). Moreover, the proposed algorithm can also be considered as generalization form of the 2point and 3point nonstationary schemes presented by Daniel and Shunmugaraj [6].
The paper is organized as follows, in section the basic notion and definitions of binary subdivision scheme are considered. The algorithm, to produce point binary nonstationary scheme, is presented in section . Some example are considered, to construct the masks of 2point, 3point and 4point schemes, in section 4. The convergence and smoothness of the schemes are being calculated in section . Some properties and advantages of proposed algorithm are being discussed in section . The conclusion is drawn in section .
2 Preliminaries
In univariate subdivision scheme, following the notion and definitions introduced in [10], the set of control points of polygon at level is mapped to a refined polygon to generate the new set of control points at the level by applying the following repeated application of the refinement rule
( 2. 1 ) 
with the sequence of finite sets of real coefficients constitute the socalled mask of subdivision scheme. If the mask of a scheme are independent of , namely if for all then it is called stationary , otherwise it is called nonstationary .
The convergent subdivision scheme, formally denoted by , with the corresponding mask necessarily satisfies
A binary nonstationary subdivision scheme is said to be convergent if for every initial data there exits a continuous limit function such that
and is not identically zero for some initial data .
We also recall that two univariate binary schemes and are said to be asymptotically equivalent if
where
Theorem 2.1. The nonstationary scheme and stationary scheme are said to be asymptotically equivalent schemes, if they have finite masks of the same support. The stationary scheme is and
then nonstationary scheme is also said to be .
3 The algorithm for approximating schemes
In this section, an algorithm has been introduced to produce
point binary nonstationary subdivision schemes (for any
integer ) which can generate the families of
limiting curves by choosing a tension parameter .
The algorithm has been established using uniform trigonometric
Bsplines of order n. So, in view of Koch et
al. [11], trigonometric Bsplines can be defined as
follow. Let and , then Uniform
Trigonometric Bsplines of
order associated with the knot sequence with the mesh size are
defined by the recurrence relation,
for
( 3. 2 ) 
and for . The trigonometric Bspline is supported on and it is the interior of its support. Moreover, are linearly independent set on the interval . Hence, on this interval, any uniform trigonometric spline has a unique representation of the form .
To obtain the mask we use the following recreance relation, for any value of ,
( 3. 3 ) 
where , with mesh size , is a trigonometric Bspline basis function of order and can be calculated form equation (3.2). It can also be observed that the schemes produced by the proposed algorithm do not have the convex hull and affine invariance properties. Since the sums of the weights of the obtained schemes at level are not equal to unity. To get sum of the mask equal to unity, the corresponding normalized scheme can be obtained (see [6]). In the following, some examples are considered to produce the masks of 2point, 3point and 4point binary approximating schemes after setting , 3 and 4, respectively, in above recurrence relation.
4 Construction of the schemes
In this section, some applications are considered to construct the masks of 2point, 3point and 4point approximating nonstationary schemes.
4.1 The 2point Approximating Scheme
The linear trigonometric Bspline basis function , with mesh size , can be calculated by setting in relation (3.3). The 2point binary nonstationary scheme (which is also called corner cutting scheme) with mask is defined, for any value of and , as
( 4. 4 ) 
where
Theorem 4.1.1. The 2point binary nonstationary scheme defined above converges and has smoothness, for the range .
Proof. see [6].
Remark 4.1.2. It can be observed that the mask of above 2point normalized scheme converges to the mask of the famous corner cutting scheme introduced by Chaikin [3]. Moreover,
it can also be observed that the mask of binary 2point nonstationary scheme of Daniel and Shummugaraj [6] can be calculated, after setting in (4). Hence the proposed
scheme can be considered as the generalized form of nonstationary scheme presented by Daniel and Shummugaraj and nonstationary counterpart of famous Chaikin’s scheme [3].
4.2 The 3point Approximating Scheme
To get quadratic trigonometric Bspline basis function, with mesh size , we take in equation (3.3). The masks of the proposed binary 3point scheme can be calculated from quadratic trigonometric Bspline function.
The 3point nonstationary scheme is defined, for some value of , as follow
( 4. 5 ) 
where
Theorem 4.1.3. The 3point binary nonstationary scheme defined above converges and has smoothness, for the range .
Proof. see [6].
Remark 4.1.4. The proposed scheme (4.5) is considered as the generalized
form of the nonstationary 3point scheme developed by Daniel and Shummugaraj in [6].
Furthermore, the proposed 3point scheme (4.5) can be considered as
the nonstationary counterpart of the stationary scheme [14].
4.3 The 4point approximating scheme
In this section, a 4point binary approximating nonstationary subdivision scheme is presented and masks of the proposed 4point binary scheme can be calculated, for any value of , using the relation (4). Where , with mesh size , is called the cubic trigonometric Bspline basis functions and can be calculated from the recurrence relation (3). The proposed scheme is defined, for some value of , as
( 4. 6 ) 
where
the proposed 4point scheme can also be considered as the general form of stationary 4point binary approximating scheme, which was introduced by Siddiqi and Ahmad [9]. The subdivision rules to refine the control polygon are defined as
( 4. 7 ) 
As the weights of the mask of the proposed scheme (4.6) are bounded by the coefficient of the mask of the above scheme (4.7). So, we can write as
The proofs of , , and can be followed from the lemma (5.1.1).
5 Convergence Analysis
The theory of asymptotic equivalence is used to investigate the convergence and smoothness of the proposed scheme following [8]. Some estimations of are bring into play to prove the convergence of the proposed schemes. To establish the estimations some inequalities are being considered.
and
5.1 Convergence Analysis of 4point Scheme
To prove the convergence and smoothness of scheme (4.6),
estimations of
are being calculated in the following lemmas.
Lemma 5.1.1. For and
Proof. To prove the inequality (i)
and
The proofs of (ii), (iii) and (iv) can be
obtained similarly.
Lemma 5.1.2. For some constants and
independent of , we have
Proof. To prove the inequality (i) use Lemma (5.1.1),
The proofs of (ii), (iii) and (iv) can be obtained similarly.
Lemma 5.1.3. The laurent polynomial
of the scheme at the level can be written as
, where
Proof. Since,
Therefore using
, it can be proved.
Lemma 5.1.4. The laurent polynomial of
the scheme at the level can be written as , where
Proof. To prove that the subdivision scheme corresponding to the symbol is , we have
Since the norm of the subdivision scheme is
So in view of Dyn [10], the stationary scheme is .
Theorem 5.1.5. The 4point nonstationary scheme defined in Eq. (4.6) converges and has smoothness, for the range .
Proof. To prove the proposed scheme to be , it
is sufficient to show that the scheme corresponding to the symbol
is (in view of Theorem given by Dyn and Levin [8].
Since is by
Lemma (5.1.4). So, it is sufficient to show for the convergence of
binary nonstationary scheme (4.6) that,
Where,
Following Lemmas and , we have
and similarly, it may be noted that
From (i), (ii), (iii) and (iv) of lemma (5.1.2), , , and . Hence, it can be written as
Thus by the Theorem 2.1, is as the
associated scheme is . Hence, proposed scheme is .
6 Properties and advantages of algorithm
In this section, some properties like unit circle reproduction property, symmetry of basis function and some other advantages of the proposed algorithm are being considered.
6.1 Reproduction of unit circle
It can be observed that certain functions like and can be reproduced
by the proposed nonstationary schemes. In particular, if a set of equidistant
point and , then the limit curve
is the unit circle (see also figure 2). Similarly, reproduces .
Proposition 6.1. The limit curves of the scheme (4.4)
reproduces the functions and
for the data points and , respectively.
In other wards we have to show that for
(i) , we have for as
(ii) Similarly, for , we have for as
Proof: For any initial data of the form , it can be followed
Similarly, it can be followed for refinement
Analogously, it can also be proved.
Proof of part (ii) is similar. So, the scheme (4.4) can
reproduce for the initial data
.
Proposition 6.2. The limit curves of the scheme (4.5)
reproduces the functions and
for the data points and , respectively.
It can be proceed on same way. If a set of equidistant point
and
can be chosen by the scheme (4.5), then
the limit curve is the unit circle. Similarly, reproduces .
6.2 Symmetry of Basis Limit Function
The basis limit
function of the scheme is the limit function for the data
In order to prove that the basis limit function is symmetric about the
Yaxis.
Theorem 6.1 The basis limit function F
is symmetric about the Yaxis.
Proof. The symmetry of basis limit function can be followed on the same pattern following [15].
6.3 Special cases
It can be observed that the binary subdivision schemes presented in [3, 6, 9, 13, 14, 15, 16] are either the special cases or can be considered the nonstationary counterpart of the stationary schemes. (See also Table 1 and Table 2).

The limiting curves can be obtained after taking in proposed algorithm. The obtained curves of scheme (5), taking , coincide with the limit curves of the famous corner cutting scheme of Chaikin [3].

After setting and in proposed algorithm, the mask of 2point and 3point approximating nonstationary schemes, developed by Daniel and Shunmugaraj [6], can be obtained.

The limit curves of 3point stationary approximating scheme, introduced by Siddiqi and Ahamd [14], coincide with the limit curves of proposed 3point scheme, after setting and .

The limit curves of proposed 4point scheme coincide with the limit curves obtained by the scheme [9], for and .

The limit curves of 5point stationary approximating scheme introduced in [16] matched with the limit curves of proposed scheme, for setting and .

The limit curves of 6point stationary approximating scheme introduced in [13] matched with the limit curves of proposed scheme, for setting and .

The proposed algorithm can also be considered as the nonstationary counterpart of point scheme developed by Siddiqi and Younis [15].
Setting  Scheme  Type  Continuity  Counterpart 

2  2point  Stationary  scheme [3]  
3  3point  Stationary  scheme [14]  
4  4point  Stationary  scheme [9]  
5  5point  Stationary  scheme [16]  
6  6point  Stationary  scheme [13]  
point  Stationary  scheme [15] 
7 Conclusion
An algorithm of point binary approximating nonstationary subdivision scheme (for any integer ) has been developed which generates the family of limiting curve, for . The construction of the algorithm is associated with trigonometric Bspline basis function. It is also evident from the examples that the limit curves of the proposed schemes coincide with the schemes presented in [3, 6, 14, 9, 13, 16]. So, the proposed algorithm, for different values of , can be considered as the generalized form of the scheme [6] and nonstationary counterpart of the stationary schemes [3, 9, 13, 14, 16]. Moreover, the schemes produced by the algorithm can reproduce or regenerate the trigonometric polynomials, trigonometric splines and conic sections as well.
References
 C. Beccari, G. Casciola and L. Romani, A nonstationary uniform tension controlled interpolating 4point scheme reproducing conics, Comput. Aided. Geom. D. , 24(1) (2007), 1–9.
 C. Beccari, G. Casciola and L. Romani, An interpolating 4point ternary non stationary subdivision scheme with tension control, Comput. Aided. Geom. D. , 24(4) (2007), 210–219.
 G.M. Chaikin, An algorithm for high speed curve generation, Comput. Vision. Graph., 3(4) (1974), 346–349.
 C. Conti and L. Romani, A new family of interpolatory nonstationary subdivision schemes for curve design in geometric modeling,in: Numerical Analysis and Applied Mathematics, International Conference VolI (2010).
 C. Conti and L. Romani, Algebraic conditions on nonstationary subdivision symbols for exponential polynomial reproduction, J. Comput. Appl. Math. , 236 (2011), 543–556.
 S. Daniel and P. Shunmugaraj, An approximating nonstationary subdivision scheme, Comput. Aided. Geom. D , 26 (2009), 810–821.
 N. Dyn, J.A. Gregory and D. Levin, A 4points interpolatory subdivision scheme for curve design, Comput. Aided. Geom. D. , 4(4) (1987), 257–268.
 N. Dyn and D. Levin, Analysis of asymptotically equivalent binary subdivision schemes, J. Math. Anal. Appl. , 193 (1995), 594–621.
 S.S. Siddiqi and N. Ahmad, An approximating stationary subdivision scheme, Eur. J. Sci. Res., 15(1) (2006), 97–102.
 N. Dyn and D. Levin, Subdivision schemes in geometric modeling, Acta Numerica , 11 (2002), 73–144.
 P.E. Koch, T. Lyche, M. Neamtu and L. Schumker, Control curves and knot insertion for trignometric splines, Adv. Comput. Math. , 3 (1995), 405–424.
 J. Pana, S. Lin and X. Luo, A combined approximating and interpolating subdivision scheme with continuity, Appl. Math. Lett., 25(12) (2012), 2140–2146.
 S.S. Siddiqi and N. Ahmad, A approximating subdivision scheme, Appl. Math. Lett., 21(7) (2008), 722–728.
 S.S. Siddiqi and N. Ahmad, A new threepoint approximating subdivision scheme, Appl. Math. Lett., 20(6) (2007), 707–711.
 S.S. Siddiqi and M. Younis, Construction of mpoint binary approximating subdivision schemes, Appl. Math. Lett. (2012), doi:10.1016/j.aml.2012.09.016.
 S.S. Siddiqi and N. Ahmad, A new fivepoint approximating subdivision scheme, Int. J. Comput. Math., 85(1) (2008), 65–72.