Piecewise-quadratics and exponential parameterization for reduced data

Abstract

In this paper we discuss the problem of interpolating the so-called reduced dataQm={qi}i=0m to estimate the unknown curve γ satisfying γ(ti)=qi. The corresponding interpolation knots {ti}i=0m for the reduced data are assumed to be unknown. The main issue for such non-parametric data fitting (given selected interpolation scheme) is to complement {ti}i=0m with somehow guessed {tˆi}i=0m, so that the respective trajectory estimation is achieved with possibly fast orders. This work discusses the so-called exponential parameterizations used in computer graphics for curve modeling, which is combined here with piecewise-quadraticsγˆ2. Such family of guessed knots {tˆiλ}i=0m (with 0⩽λ⩽1) comprises well-known cases. Indeed, for λ=0a blind uniform guess follows. When λ=1/2 the so-called centripetal parameterization occurs. On the other hand, if λ=1cumulative chords are invoked. As already established for γˆ2, cumulative chords for all admissible samplings match a cubic convergence order α(1)=3 holding also for the corresponding non-reduced data({ti}i=0m,Qm) used with piecewise-quadratics γ̃2 (i.e. a special case of the so-called parametric interpolation). Our main result reads that, for exponential parameterization and curves sampled more-or-less uniformly (forming a wide subclass of admissible samplings) the resulting convergence rates α(λ) for λ∈[0,1) do not accelerate continuously from α(0)=1 to α(1)=3. In fact, they all coincide and satisfy sharp condition α(λ)=α(0)=1. The sharpness of the asymptotic estimates between curve γ and its interpolant γˆ2 derived herein is confirmed by theoretical argument and independently verified by experimental testing. In addition, we also demonstrate that, a natural candidate for re-parameterization of γˆ2 to synchronize both domains of γ and γˆ2 (necessary to derive the asymptotics in question) may fail to form a genuine re-parameterization.