New spline basis functions for sampling approximations

Ueno, Toshihide; Truscott, Simon; Okada, Masami

doi:10.1007/s11075-007-9119-x

Cited by 4 publications

(21 citation statements)

References 8 publications

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“…(a) if c k = 0 (k = 1, 2, · · · , n) then the spline ϕ 2n (x) coincides with one, constructed in [5].…”

Section: In Particularmentioning

confidence: 99%

“…In [5] the spline basis functions ϕ 2n (x) which are symmetric with narrower compact support are constructed satisfying the following conditions (C1) ϕ 2n (−x) = ϕ 2n (x), (C2) suppϕ 2n (x) = [−n − 1, n + 1], (C3) ϕ 2n (k) = δ k,0 , (k ∈ Z), (C4) k∈Z k i ϕ 2n (x − k) = x i , i = 0, 1, · · · , 2n.…”

Section: Introductionmentioning

confidence: 99%

“…In [5] it is also shown that if ϕ 2n (x) is a specific linear combination of {N 2n+2 , · · · , N n+2 } and satisfies (C3), it enjoys the condition (C4) which is equivalent to the moment condition. As mentioned in [5], the construction of such basis functions is interesting from both the sampling and interpolating approximation point of view.…”

Section: Introductionmentioning

confidence: 99%

“…As mentioned in [5], the construction of such basis functions is interesting from both the sampling and interpolating approximation point of view. But the general rule for choosing the specific linear combination mentioned above is complicated.…”

Section: Introductionmentioning

confidence: 99%

“…

Abstract: A method for constructing a new kind of spline basis functions (ϕ2n(x)) with compact support on R was described by Ueno et al (2007). These basis functions for sampling approximations consist of a linear combination of the cardinal B splines, but the construction is complicated and therefore were constructed basis functions ϕ2n(x) only for n ≤ 5.

…”

mentioning

confidence: 99%

See 4 more Smart Citations

A New Method for the Construction of Spline Basis Functions for Sampling Approximations

Жанлав¹,

Mijiddorj²

2016

Int. J. of Pure and Appl. Math.

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A method for constructing a new kind of spline basis functions (ϕ2n(x)) with compact support on R was described by Ueno et al. (2007). These basis functions for sampling approximations consist of a linear combination of the cardinal B splines, but the construction is complicated and therefore were constructed basis functions ϕ2n(x) only for n ≤ 5. We discuss a new construction of the basis functions and its approximation properties are considered.

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“…(a) if c k = 0 (k = 1, 2, · · · , n) then the spline ϕ 2n (x) coincides with one, constructed in [5].…”

Section: In Particularmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

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A New Method for the Construction of Spline Basis Functions for Sampling Approximations

Жанлав¹,

Mijiddorj²

2016

Int. J. of Pure and Appl. Math.

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Interpolatory blending net subdivision schemes of Dubuc–Deslauriers type

Conti

Dyn

Romani

2012

Computer Aided Geometric Design

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Net subdivision schemes recursively refine nets of univariate continuous functions defined on the lines of planar grids, and generate as limits bivariate continuous functions. In this paper a family of interpolatory net subdivision schemes related to the family of Dubuc-Deslauriers interpolatory subdivision schemes is constructed and analyzed. The construction is based on Gordon blending interpolants to nets of univariate functions, and on a particular class of blending functions with properties related to the Dubuc-Deslauriers schemes. The general analysis tools for net subdivision schemes, developed in a previous paper by the authors, together with the properties of the blending functions, lead to the proof of the convergence of these schemes to limit functions having the same integer smoothness as the limits of the corresponding Dubuc-Deslauriers schemes. These results are proved for net subdivision schemes corresponding to the first 84 members of the Dubuc-Deslauriers family, and conjectured for the rest. A concrete example of a family of piecewise polynomial blending functions is considered, together with the corresponding family of net subdivision schemes. The performance of the first two net subdivision schemes in this family is demonstrated by two examples.

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High quality local interpolation by composite parametric surfaces

Antonelli

Beccari

Casciola

2016

Computer Aided Geometric Design

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In CAGD the design of a surface that interpolates an arbitrary quadrilateral mesh is definitely a challenging task. The basic requirement is to satisfy both criteria concerning the regularity of the surface and aesthetic concepts.With regard to the aesthetic quality, it is well known that interpolatory methods often produce shape artifacts when the data points are unevenly spaced. In the univariate setting, this problem can be overcome, or at least mitigated, by exploiting a proper non-uniform parametrization, that accounts for the geometry of the data. Moreover, recently, the same principle has been generalized and proven to be effective in the context of bivariate interpolatory subdivision schemes.In this paper, we propose a construction for parametric surfaces of good aesthetic quality and high smoothness that interpolate quadrilateral meshes of arbitrary topology.In the classical tensor product setting the same parameter interval must be shared by an entire row or column of mesh edges. Conversely, in this paper, we assign a different parameter interval to each edge of the mesh. This particular structure, which we call an augmented parametrization, allows us to interpolate each section polyline of the mesh at parameters values that prevent wiggling of the resulting curve or other interpolation artifacts. This yields high quality interpolatory surfaces.The proposed surfaces are a generalization of the local univariate spline interpolants introduced in Beccari et al. (2013) and Antonelli et al. (2014), that can have arbitrary continuity and arbitrary order of polynomial reproduction. In particular, these surfaces retain the same smoothness of the underlying class of univariate splines in the regular regions of the mesh (where, locally, all vertices have valence 4). Moreover, in mesh regions containing vertices of valence other than 4, we suitably define G 1 -or G 2 -continuous surface patches that join the neighboring regular ones.

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New spline basis functions for sampling approximations

Cited by 4 publications

References 8 publications

A New Method for the Construction of Spline Basis Functions for Sampling Approximations

A New Method for the Construction of Spline Basis Functions for Sampling Approximations

Interpolatory blending net subdivision schemes of Dubuc–Deslauriers type

High quality local interpolation by composite parametric surfaces

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