An approximating non-stationary subdivision scheme

Daniel, Sunita; Shunmugaraj, P.

doi:10.1016/j.cagd.2009.02.007

Cited by 31 publications

(34 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As discussed in Section 2.3 (see Remark 2.2), if compared with the corresponding order-3 NULIFS interpolant, the limit curve of the NULI 4-point scheme turns out to be tighter to the initial data polygon (in the sense of Remark 2.2). Although there are many proposals of stationary and non-stationary subdivision schemes whose refinement equations include shape parameters [1][2][3]12,17,19,30], the authors are not aware of any existing scheme whose parameters set has a behavior comparable to the NULI 4-point scheme. In fact, so far parameters have been introduced either to control the tension of the limit curve [2,3,30], to increase its smoothness [17,19] or to reproduce salient curves [1,3,12,30].…”

Section: Application Examples and Comparisonsmentioning

confidence: 99%

Non-uniform interpolatory curve subdivision with edge parameters built upon compactly supported fundamental splines

2011

View full text Add to dashboard Cite

In this paper we present a family of Non-Uniform Local Interpolatory (NULI) subdivision schemes, derived from compactly supported interpolatory fundamental splines with non-uniform knots (NULIFS). For this spline family, the knot-partition is defined by a sequence of break points and by one additional knot, arbitrarily placed along each knot-interval. The resulting refinement algorithms are linear and turn out to contain a set of edge parameters that, when fixed to a value in the range [0,1], allow us to achieve special shape features by simply moving each auxiliary knot between the break points. Among all the members of this new family of schemes, we will then especially analyze the NULI 4-point refinement. This subdivision scheme has all the fundamental features of the quadratic fundamental spline basis it is originated from, namely compact support, C 1 smoothness, second order polynomials reproduction and approximation order 3. In addition the NULI 4-point subdivision algorithm has the possibility of setting consecutive edge parameters to simulate double and triple knots -that are not considered by the authors of the corresponding spline basis -thus allowing for limit curves with crease vertices, without using an ad hoc mask. Numerical examples and comparisons with other methods will be given to the aim of illustrating the performance of the NULI 4-point scheme in the case of highly non-uniform initial data.

show abstract

Section: Application Examples and Comparisonsmentioning

confidence: 99%

Non-uniform interpolatory curve subdivision with edge parameters built upon compactly supported fundamental splines

2011

View full text Add to dashboard Cite

show abstract

“…) and the masks of above scheme are bounded by the masks given in (17). The subdivision scheme defined by Ko et al [13], to refine the control polygon, is given below Figure 2.…”

Section: The 4-point Schemementioning

confidence: 99%

“…Jena et al [8] introduced a 4-point binary interpolatory non-stationary C 1 subdivision scheme which was the generalization of four point stationary subdivision scheme developed by Dyn et al [11]. In 2009, Daniel and Shunmugaraj introduced a non-stationary 2-points approximating scheme that generates C 1 limiting curve and two 3-points schemes that generate C 2 and C 3 limiting curves using trigonometric B-spline basis function [17]. Lee and Yoon [19] introduced non-stationary subdivision schemes for surface interpolation based on exponential polynomials that can reproduce a large class of (complex) exponential polynomials.…”

Section: Introductionmentioning

confidence: 99%

Ternary approximating non-stationary subdivision schemes for curve design

Siddiqi

Younis²

2014

Open Engineering

View full text Add to dashboard Cite

Abstract:In this paper, an algorithm has been introduced to produce ternary 2m-point (for any integer m ≥ 1) approximating non-stationary subdivision schemes which can generate the linear spaces spanned by {1; cos(α.); sin(α.)}. The theory of asymptotic equivalence is being used to analyze the convergence and smoothness of the schemes. The proposed algorithm can be consider as the non-stationary counter part of the 2-point and 4-point existing ternary stationary approximating schemes, for different values of m. Moreover, the proposed algorithm has the ability to reproduce or regenerate the conic sections, trigonometric polynomials and trigonometric splines.

show abstract

“…Moreover, {T n j } m j=1 are linearly independent sets of the interval [t n−1 , t m+1 ]. Hence, on [t n−1 , t m+1 ], any uniform trigonometric spline S(x) has a unique representation of the form S(x) = m j=0 p j T n j (x; α), p j ∈ R; see also [7].…”

Section: Preliminariesmentioning

confidence: 99%

“…A 4-point ternary interpolating non-stationary subdivision scheme that generates C 2 continuous limit curves, showing considerable variation of shapes with a tension parameter, was presented by the same authors in the same year [2]. In 2009, Daniel and Shunmugaraj [7] developed a non-stationary 2-point approximating scheme that generates C 1 limiting curves and two 3-point binary schemes that generate C 2 and C 3 limiting curves. The masks of these schemes were defined in terms of trigonometric B-spline basis functions.…”

Section: Introductionmentioning

confidence: 99%

A symmetric non-stationary subdivision scheme

Siddiqi

Younis

2014

LMS J. Comput. Math.

View full text Add to dashboard Cite

This paper proposes a new family of symmetric 4-point ternary non-stationary subdivision schemes that can generate the limit curves of C 3 continuity. The continuity of this scheme is higher than the existing 4-point ternary approximating schemes. The proposed scheme has been developed using trigonometric B-spline basis functions and analyzed using the theory of asymptotic equivalence. It has the ability to reproduce or regenerate the conic sections, trigonometric polynomials and trigonometric splines as well. Some graphical and numerical examples are being considered, by choosing an appropriate tension parameter 0 < α < π/3, to show the usefulness of the proposed scheme. Moreover, the Hölder regularity and the reproduction property are also being calculated.

show abstract

An approximating non-stationary subdivision scheme

Cited by 31 publications

References 14 publications

Non-uniform interpolatory curve subdivision with edge parameters built upon compactly supported fundamental splines

Non-uniform interpolatory curve subdivision with edge parameters built upon compactly supported fundamental splines

Ternary approximating non-stationary subdivision schemes for curve design

A symmetric non-stationary subdivision scheme

Contact Info

Product

Resources

About