Abstract
This paper tackles the problem of interpolating reduced data \(Q_m=\{q_i\}_{i=0}^m\) obtained by sampling an unknown curve γ in arbitrary euclidean space. The interpolation knots \({\cal T}_m= \{t_i\}_{i=0}^m\) satisfying γ(t i ) = q i are assumed to be unknown (non-parametric interpolation). Upon selecting a specific numerical scheme \(\hat \gamma\) (here a piecewise-quadratic \(\hat \gamma=\hat \gamma_2\)), one needs to supplement Q m with knots’ estimates \(\{\hat t_i\}_{i=0}^m\approx\{t_i\}_{i=0}^m\). A common choice of \(\{\hat t_i^{\lambda}\}_{i=0}^m\) (λ ∈ [0,1]) frequently used in curve modeling and data fitting (e.g. in computer graphics and vision or in computer aided design) is called exponential parameterizations (see, e.g., [11] or [16]). Recent results in [8]and [14] show that \(\hat \gamma_2\) combined with exponential parameterization yields (in trajectory estimation) either linear α(λ) = 1 (λ ∈ [0,1)) or cubic α(1) = 3 convergence orders, once Q m gets progressively denser. The asymtototics proved in [8] relies on the extra assumptions requiring \(\hat \gamma_2\) to be reparameterizable to the domain of γ. Indeed, as shown in [14], a natural candidate ψ for such a reparameterization meets this criterion only for λ = 1, whereas the latter (see [8]) may not hold for the remaining λ ∈ [0,1) (which e.g. brings difficulty in length estimation of γ by using \(\hat \gamma\)). Our paper fills out this gap and establishes sufficient conditions imposed on \({\cal T}_m\) to render ψ a genuine reparameterization with λ ∈ [0,1) (see Th. 4). The derivation of a such a condition involves theoretical analysis and symbolic computation, and this constitutes a novel contribution of the present work. The numerical tests verifying whether ψ indeed is a reparameterization (for λ ∈ [0,1) and for more-or-less uniform samplings \({\cal T}_m\)) are also performed. The sharpness of the asymptotics in question is additionally confirmed with the aid of numerical tests.
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Kozera, R., Noakes, L. (2015). Piecewise-Quadratics and Reparameterizations for Interpolating Reduced Data. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2015. Lecture Notes in Computer Science(), vol 9301. Springer, Cham. https://doi.org/10.1007/978-3-319-24021-3_20
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