Convergence order in trajectory estimation by piecewise-cubics and exponential parameterization

Ryszard Kozera, Magdalena Wilkołlazka

Research output: Contribution to journalArticle

Abstract

This paper disusses the problem of estimating the trajetory of the unknown urve γ from the sequene of m + 1 interpolation points Qm = {γ(ti)}m i=0 in arbitrary Eulidean spae En. The respetive knots Tm = {ti}m i=0 (in asending order) are assumed to be unknown. Suh Qm isT oined reduced data. In our setting, a pieewise-ubi Lagrange interpolation γ3: [0, ] →T En is applied to fit Qm. Here, the missing knots Tm are replaed by their estimatesm = {ti}m i=0 in aordane with the exponential parameterization. The latter is ontrolled by a single parameter λ ∈ [0, 1]. This work analyzes the intrinsi asymptotis in approximating γ by γ3 based on the exponential parameterization and Qm. The multiple goals are ahieved. Firstly, the existing result established for λ = 1 (i.e. for the umulative hord parameterization) is extended to the remaining ases of λ ∈ [0, 1) and more-or-less uniformly sampled Qm. As demonstrated herein, a quarti onvergene order α(1) = 4 in trajetory estimation drops disontinuously to the linear one α(λ) = 1, for all λ ∈ [0, 1). Seondly, the asymptotis derived in this paper is also analytially proved to be sharp with the aid of illustrative examples. Thirdly, the latter is verified in affirmative upon onduting numerial testing. Next, the neessity of more-or-less uniformity imposed on Qm is shown to be indispensable. In addition, several suffiient onditions for γ3 to be reparameterizable to the domain of γ are formulated. Lastly, the motivation for using the exponential parameterization with λ ∈ [0, 1) is also outlined.

Original languageEnglish
Pages (from-to)72-94
Number of pages23
JournalMathematical Modelling and Analysis
Volume24
Issue number1
DOIs
Publication statusPublished - 21 Nov 2019

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Convergence Order
Parameterization
Trajectories
Trajectory
Knot
Interpolation
Unknown
Lagrange Interpolation
Uniformity
Interpolate
Testing
Arbitrary

Cite this

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abstract = "This paper disusses the problem of estimating the trajetory of the unknown urve γ from the sequene of m + 1 interpolation points Qm = {γ(ti)}m i=0 in arbitrary Eulidean spae En. The respetive knots Tm = {ti}m i=0 (in asending order) are assumed to be unknown. Suh Qm isT oined reduced data. In our setting, a pieewise-ubi Lagrange interpolation γ3: [0, ] →T En is applied to fit Qm. Here, the missing knots Tm are replaed by their estimatesm = {ti}m i=0 in aordane with the exponential parameterization. The latter is ontrolled by a single parameter λ ∈ [0, 1]. This work analyzes the intrinsi asymptotis in approximating γ by γ3 based on the exponential parameterization and Qm. The multiple goals are ahieved. Firstly, the existing result established for λ = 1 (i.e. for the umulative hord parameterization) is extended to the remaining ases of λ ∈ [0, 1) and more-or-less uniformly sampled Qm. As demonstrated herein, a quarti onvergene order α(1) = 4 in trajetory estimation drops disontinuously to the linear one α(λ) = 1, for all λ ∈ [0, 1). Seondly, the asymptotis derived in this paper is also analytially proved to be sharp with the aid of illustrative examples. Thirdly, the latter is verified in affirmative upon onduting numerial testing. Next, the neessity of more-or-less uniformity imposed on Qm is shown to be indispensable. In addition, several suffiient onditions for γ3 to be reparameterizable to the domain of γ are formulated. Lastly, the motivation for using the exponential parameterization with λ ∈ [0, 1) is also outlined.",
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Convergence order in trajectory estimation by piecewise-cubics and exponential parameterization. / Kozera, Ryszard; Wilkołlazka, Magdalena.

In: Mathematical Modelling and Analysis, Vol. 24, No. 1, 21.11.2019, p. 72-94.

Research output: Contribution to journalArticle

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