A generalized curve subdivision scheme of arbitrary order with a tension parameter

Fang, Meie; Ma, Weiyin; Wang, Guozhao

doi:10.1016/j.cagd.2010.09.001

Cited by 32 publications

(19 citation statements)

References 17 publications

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“…The interest in nonstationary subdivision schemes arose in the last ten years after it was pointed out that they can be equipped with tension parameters that allow us to get as close as desired to the original mesh and to obtain considerable variations of shape (see [16,30,39,40,42,43]). Indeed, differently from stationary subdivision schemes, nonstationary subdivision schemes are capable of reproducing conic sections, spirals or, in general, of generating exponential polynomials x r e θx , x ∈ R, r ∈ N ∪ {0}, θ ∈ C. This generation property is important not only in geometric design (see, e.g., [30,32,37,42,43]), but also in many other applications, e.g., in biomedical imaging (see, e.g., [20,21]) and in Isogeometric Analysis (see, e.g., [19,31]).…”

Section: Non-stationary Subdivision Schemes and Exponential Polynomiamentioning

confidence: 99%

“…Indeed, differently from stationary subdivision schemes, nonstationary subdivision schemes are capable of reproducing conic sections, spirals or, in general, of generating exponential polynomials x r e θx , x ∈ R, r ∈ N ∪ {0}, θ ∈ C. This generation property is important not only in geometric design (see, e.g., [30,32,37,42,43]), but also in many other applications, e.g., in biomedical imaging (see, e.g., [20,21]) and in Isogeometric Analysis (see, e.g., [19,31]). However, the use of nonstationary subdivision schemes in IgA is nowadays limited to the case of exponential B-splines since they are the only functions that have been shown to be able to overcome the NURBS limits while preserving their useful properties.…”

Section: Non-stationary Subdivision Schemes and Exponential Polynomiamentioning

confidence: 99%

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Exponential pseudo-splines: Looking beyond exponential B-splines

Conti

Gemignani

Romani

2016

Journal of Mathematical Analysis and Applications

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Pseudo-splines are a rich family of functions that allows the user to meet various demands for balancing polynomial reproduction (i.e., approximation power), regularity and support size. Such a family includes, as special members, B-spline functions, universally known for their usefulness in different fields of application. When replacing polynomial reproduction by exponential polynomial reproduction, a new family of functions is obtained. This new family is here constructed and called the family of exponential pseudo-splines. It is the nonstationary counterpart of (polynomial) pseudo-splines and includes exponential B-splines as a special subclass. In this work we provide a computational strategy for deriving the explicit expression of the Laurent polynomial sequence that identifies the family of exponential pseudo-spline nonstationary subdivision schemes. For this family we study its symmetry properties and perform its convergence and regularity analysis. Finally, we also show that the family of primal exponential pseudo-splines fills in the gap between exponential B-splines and interpolatory cardinal functions. This extends the analogous property of primal pseudo-spline stationary subdivision schemes.

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Section: Non-stationary Subdivision Schemes and Exponential Polynomiamentioning

confidence: 99%

Section: Non-stationary Subdivision Schemes and Exponential Polynomiamentioning

confidence: 99%

Exponential pseudo-splines: Looking beyond exponential B-splines

Conti

Gemignani

Romani

2016

Journal of Mathematical Analysis and Applications

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“…Similar methods are also developed for subdivision schemes on triangular meshes with 1-4 splitting [Stam 2001] and √ 3-subdivision [Oswald and Schröder 2003]. Recently, local refinement rules have also been presented for nonuniform B-spline curves [Cashman et al 2009b;Schaefer and Goldman 2009], nonuniform B-spline surfaces [Cashman et al 2009a], and a generalized subdivision scheme that reproduces classic B-spline, trigonometric B-spline, and hyperbolic B-spline curves [Fang et al 2010], all of arbitrary degree.…”

Section: Introductionmentioning

confidence: 95%

A unified interpolatory subdivision scheme for quadrilateral meshes

Deng

2013

ACM Trans. Graph.

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For approximating subdivision schemes, there are several unified frameworks for effectively constructing subdivision surfaces generalizing splines of an arbitrary degree. In this article, we present a similar unified framework for interpolatory subdivision schemes. We first decompose the 2n-point interpolatory curve subdivision scheme into repeated local operations. By extending the repeated local operations to quadrilateral meshes, an efficient algorithm can be further derived for interpolatory surface subdivision. Depending on the number n of repeated local operations, the continuity of the limit curve or surface can be of an arbitrary order C L , except in the surface case at a limited number of extraordinary vertices where C 1 continuity with bounded curvature is obtained. Boundary rules built upon repeated local operations are also presented. ACM Reference Format:Deng, C. and Ma, W. 2013. A unified interpolatory subdivision scheme for quadrilateral meshes.

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“…The outline of this paper is organized as follows. Section 2 provides a short survey of required background, including a summary on the stationary linear generalized subdivision scheme of arbitrary degree with a tension parameter introduced in [10] and a brief review on the circle average. In Sect.…”

Section: Introductionmentioning

confidence: 99%

A nonlinear generalized subdivision scheme of arbitrary degree with a tension parameter

et al. 2020

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In this paper, a nonlinear generalized subdivision scheme of arbitrary degree with a tension parameter is presented, which refines 2D point-normal pairs. The construction of the scheme is built upon the stationary linear generalized subdivision scheme of arbitrary degree with a tension parameter, by replacing the weighted binary arithmetic average in the linear scheme with the circle average. For such a nonlinear scheme, we investigate its smoothness and get that it can reach $C^{1}$ C 1 with suitable choices of the tension parameter when degree $m\geq 3$ m ≥ 3 . Besides, the nonlinear scheme can reconstruct the circle and the selection of parameters and initial normal vectors can effectively control the shape of the limit curves.

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A generalized curve subdivision scheme of arbitrary order with a tension parameter

Cited by 32 publications

References 17 publications

Exponential pseudo-splines: Looking beyond exponential B-splines

Exponential pseudo-splines: Looking beyond exponential B-splines

A unified interpolatory subdivision scheme for quadrilateral meshes

A nonlinear generalized subdivision scheme of arbitrary degree with a tension parameter

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