2013
DOI: 10.1007/s10444-013-9335-y
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A general framework for the construction of piecewise-polynomial local interpolants of minimum degree

Abstract: In this paper we consider the problem of designing piecewise polynomial local interpolants of non-uniformly spaced data. We provide a constructive approach that, for any assigned degree of polynomial reproduction, continuity order, and support width, allows for generating the fundamental spline functions of minimum degree having the desired properties. Finally, the proposed construction is extended to handle open sets of data and to the case of multiple knots

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Cited by 16 publications
(33 citation statements)
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“…where the coefficients r[k] = (r x [k], r y [k], r z [k]) with k ∈ Z are the control points. The curve (29) can be locally modified by changing the position of a single control point. The shapes that r can adopt (e.g., polynomial, circular, elliptic) depend on the properties of the generator.…”
Section: Reproduction Of Idealized Shapesmentioning
confidence: 99%
See 1 more Smart Citation
“…where the coefficients r[k] = (r x [k], r y [k], r z [k]) with k ∈ Z are the control points. The curve (29) can be locally modified by changing the position of a single control point. The shapes that r can adopt (e.g., polynomial, circular, elliptic) depend on the properties of the generator.…”
Section: Reproduction Of Idealized Shapesmentioning
confidence: 99%
“…The shapes that r can adopt (e.g., polynomial, circular, elliptic) depend on the properties of the generator. One can also extend the curve model (29) to represent separable tensorproduct surfaces. In this case, a surface σ is parameterized by u, v ∈ R as…”
Section: Reproduction Of Idealized Shapesmentioning
confidence: 99%
“…Furthermore, from [23] we know that, for α ∈ α n 0 of multiplicity q + 1, there exist sequences p n such that (12) for n = 0, . .…”
Section: Reproduction Of Exponential Polynomialsmentioning
confidence: 99%
“…Especially in 3D applications, this can be inconvenient because it is no longer intuitive to interactively modify complex shapes. More recently a method to construct piecewise polynomial interpolators has been presented in [11,12].…”
Section: Introductionmentioning
confidence: 99%
“…The primary contributions of this work are: 1) a new geometrical representation based on subdivision. A crucial aspect is the choice of the subdivision mask that determines important properties of the model such as its approximation properties, the capability of reproducing circular, elliptical, or polynomial shapes [19], as well as the possibility of being interpolatory [20], [21] or not; 2) the derivation of associated energy functions such as regionand edge-based terms; 3) the presentation of an integrated strategy where the snake is optimized in a coarse-to-fine fashion. This multiscale approach is algorithmic and inherently recursive: We increase the number of points describing the curve as the algorithm progresses to the solution; at each step, the scale of the image feature (on which the optimization is performed) is matched to the density of the point cloud.…”
Section: Introductionmentioning
confidence: 99%