The best parameterization for parametric interpolation

Kuznetsov, E. B.; Yakimovich, A. Yu.

doi:10.1016/j.cam.2005.06.040

Cited by 20 publications

(10 citation statements)

References 3 publications

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“…In the general context of spline interpolation, the influence of the parameterization on the quality of the interpolant, as well as the issue of determining the best parameterization for any given set of data, has been widely analyzed [18,21,22]. These results suggest that, computing the knotpartition according to the centripetal parameterization…”

Section: The Free Parameters Characterizing the Family Membersmentioning

confidence: 99%

“…Although similar approaches exist in the spline context as well [10,27], it is also well-known that a properly chosen parameterization helps improving the quality of the interpolant and eliminating unwanted oscillations. In fact, a suitable parameterization does not guarantee shape preservation of the initial data, but it allows us to significantly bound the global and local deviation of the resulting curve from its data polygon [18,21,22]. In the context of interpolatory subdivision, refinement algorithms with non-uniform knots were firstly introduced by Daubechies et al [13].…”

Section: Introductionmentioning

confidence: 99%

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

2011

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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.

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Section: The Free Parameters Characterizing the Family Membersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

2011

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“…in [3,13,14,17,21,22]. A similar effect of the left-hand side discontinuity in convergence rate jumping from 1 to 3 (or from 1 to 4) at λ = 1 occurs for C 0 piecewise-quadratic (piecewise-cubic) Lagrange interpolation based on exponential parameterization and Q msee [9,10,12,20].…”

Section: Discussionmentioning

confidence: 78%

A Modified Hermite Interpolation with Exponential Parameterization

Kozera

Wilkołazka

2018

Math.Comput.Sci.

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This work discusses the problem of fitting a regular curve γ based on reduced data points Q m = (q 0 , q 1 , . . . , q m ) in arbitrary Euclidean space. The corresponding interpolation knots T = (t 0 , t 1 , . . . , t m ) are assumed to be unknown. In this paper the missing knots are estimated by The main result of this paper extends the latter to the remaining cases of exponential parameterization covering all λ ∈ [0, 1). A slower linear convergence order in trajectory estimation is established for any λ ∈ [0, 1) and arbitrary more-or-less uniform sampling. The numerical tests conducted in Mathematica indicate the sharpness of the latter and confirm the necessity of more-or-less uniformity. Other interpolation schemes used to fit reduced data Q m and based on T λ together with some relevant applications are also briefly recalled in this paper.

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“…Here, we have also used formula (12) for entries of the vector L(r + 1). Thus, the coefficient vectorl of the Nordsieck method (32) is defined by formulas (18). The theorem is proved.…”

Section: Local Error Estimation and Ien Methodsmentioning

confidence: 88%

On quasi-consistent integration by Nordsieck methods

Kulikov

2009

Journal of Computational and Applied Mathematics

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MSC:65L05 65L06 65L20 65L50 Keywords:Quasi-consistent Nordsieck methods Double quasi-consistency Implicitly extended Nordsieck formulas Adaptivity Local error estimation Automatic global error control a b s t r a c t In this paper we study advantages of numerical integration by quasi-consistent Nordsieck formulas. All quasi-consistent numerical methods possess at least one important property for practical use, which has not attracted attention yet, i.e. the global error of a quasiconsistent method has the same order as its local error. This means that the usual local error control will produce a numerical solution for the prescribed accuracy requirement if the principal term of the local error dominates strongly over remaining terms. In other words, the global error control can be as cheap as the local error control in the methods under discussion.Here, we apply the above-mentioned idea to Nordsieck-Adams-Moulton methods, which are known to be quasi-consistent. Moreover, some Nordsieck-Adams-Moulton methods are even super-quasi-consistent. The latter property means that their propagation matrices annihilate two leading terms in the defect expansion of such methods. In turn, this can impose a strong relation between the local and global errors of the numerical solution and allow the global error to be controlled effectively by a local error control. We also introduce Implicitly Extended Nordsieck methods such that in some sense they form pairs of embedded formulas with their source Nordsieck methods. This facilitates the local error control in quasi-consistent Nordsieck schemes. Numerical examples presented in this paper confirm clearly the power of quasi-consistent integration in practice.

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The best parameterization for parametric interpolation

Cited by 20 publications

References 3 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

A Modified Hermite Interpolation with Exponential Parameterization

On quasi-consistent integration by Nordsieck methods

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