Shape controlled interpolatory ternary subdivision

Beccari, Carolina Vittoria; Casciola, Giulio; Romani, Lucia

doi:10.1016/j.amc.2009.06.014

Cited by 28 publications

(26 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

“…Research is continually moving toward the investigation of refinement rules able to combine desirable reproduction properties under some geometrical constraints. In particular, schemes capable of reproducing circles were proposed in Zhang (1996), Zhang and Krause (2005), Sabin and Dodgson (2004), Deng and Wang (2010), Romani (2010), and, more recently, schemes based on exponential B-splines made possible the reproduction of conic sections (Beccari et al, 2007(Beccari et al, , 2009Sunita and Shunmugaraj, 2009;Conti and Romani, 2010;Conti et al, 2011) and exponential polynomials (Dyn et al, 2008;Romani, 2009).…”

Section: Introductionmentioning

confidence: 99%

Exponential splines and minimal-support bases for curve representation

Delgado-Gonzalo

Thévenaz

Unser

2012

Computer Aided Geometric Design

View full text Add to dashboard Cite

Our interest is to characterize the spline-like integer-shift-invariant bases capable of reproducing exponential polynomial curves. We prove that any compact-support function that reproduces a subspace of the exponential polynomials can be expressed as the convolution of an exponential B-spline with a compact-support distribution. As a direct consequence of this factorization theorem, we show that the minimal-support basis functions of that subspace are linear combinations of derivatives of exponential Bsplines. These minimal-support basis functions form a natural multiscale hierarchy, which we utilize to design fast multiresolution algorithms and subdivision schemes for the representation of closed geometric curves. This makes them attractive from a computational point of view. Finally, we illustrate our scheme by constructing minimalsupport bases that reproduce ellipses and higher-order harmonic curves.

show abstract

“…In the paper, interpolatory schemes are obtained from approximating schemes and the behavior of 6-point interpolating scheme is discussed thoroughly. In the literature, non-stationary subdivision schemes have been found to be constructed using trigonometric Lagrange [7,8,10], trigonometric B-spline [11][12][13] and exponential polynomials [6,19] while no interpolatory non-stationary subdivision scheme has been found using the hyperbolic function [14,15]. In this paper, binary 6-point interpolating non-stationary subdivision scheme is constructed using the hyperbolic function which has the ability to reproduce parabolas and hyperbolas.…”

Section: Introductionmentioning

confidence: 99%