TY - JOUR

T1 - Piecewise-quadratics and exponential parameterization for reduced data

AU - Kozera, R.

AU - Noakes, Lyle

PY - 2013

Y1 - 2013

N2 - In this paper we discuss the problem of interpolating the so-called reduced data Qm={qi}i=0 m to estimate the unknown curve γ satisfying γ(ti)=qi. The corresponding interpolation knots {ti}i=0 m 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=0 m with somehow guessed {t̂i}i=0 m, 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-quadratics γ̂ 2. Such family of guessed knots {t̂iλ}i=0 m (with 0≤λ≤1) comprises well-known cases. Indeed, for λ=0 a blind uniform guess follows. When λ=1/2 the so-called centripetal parameterization occurs. On the other hand, if λ=1 cumulative chords are invoked. As already established for γ̂2, cumulative chords for all admissible samplings match a cubic convergence order α(1)=3 holding also for the corresponding non-reduced data ({t i}i=0 m,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 γ̂2 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 γ̂2 to synchronize both domains of γ and γ̂2 (necessary to derive the asymptotics in question) may fail to form a genuine re-parameterization.

AB - In this paper we discuss the problem of interpolating the so-called reduced data Qm={qi}i=0 m to estimate the unknown curve γ satisfying γ(ti)=qi. The corresponding interpolation knots {ti}i=0 m 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=0 m with somehow guessed {t̂i}i=0 m, 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-quadratics γ̂ 2. Such family of guessed knots {t̂iλ}i=0 m (with 0≤λ≤1) comprises well-known cases. Indeed, for λ=0 a blind uniform guess follows. When λ=1/2 the so-called centripetal parameterization occurs. On the other hand, if λ=1 cumulative chords are invoked. As already established for γ̂2, cumulative chords for all admissible samplings match a cubic convergence order α(1)=3 holding also for the corresponding non-reduced data ({t i}i=0 m,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 γ̂2 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 γ̂2 to synchronize both domains of γ and γ̂2 (necessary to derive the asymptotics in question) may fail to form a genuine re-parameterization.

U2 - 10.1016/j.amc.2013.06.060

DO - 10.1016/j.amc.2013.06.060

M3 - Article

VL - 221

SP - 620

EP - 638

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

SN - 0096-3003

ER -