Piecewise-Quadratics and Reparameterizations for Interpolating Reduced Data

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.