# Trajectory Estimation for Exponential Parameterization and Different Samplings

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1 Faculty of Applied Informatics and Mathematics 2 Department of Mathematics and Statistics

Abstract : This paper discusses the issue of fitting reduced data $Q m=\{q i\} {i=0}^m$ with piecewise-quadratics to estimate an unknown curve γ in Euclidean space. The interpolation knots $\{t i\} {i=0}^m$ with γti = qi are assumed to be unknown. Such non-parametric interpolation commonly appears in computer graphics and vision, engineering and physics 1. We analyze a special scheme aimed to supply the missing knots $\{\hat t i^{\lambda}\} {i=0}^m\approx\{t i\} {i=0}^m$ with λ ∈ 0,1 - the so-called exponential parameterization used in computer graphics for curve modeling. A blind uniform guess, for λ = 0 coupled with more-or-less uniform samplings yields a linear convergence order in trajectory estimation. In addition, for ε-uniform samplings ε ≥ 0 and λ = 0 an extra acceleration αε0 =  min {3,1 + 2ε} follows 2. On the other hand, for λ = 1 cumulative chords render a cubic convergence order α1 = 3 within a general class of admissible samplings 3. A recent theoretical result 4 is that for λ ∈ 0,1 and more-or-less uniform samplings, sharp orders αλ = 1 eventuate. Thus no acceleration in αλ

Autor: Ryszard Kozera - Lyle Noakes - Piotr Szmielew -

Fuente: https://hal.archives-ouvertes.fr/

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