Trajectory Estimation for Exponential Parameterization and Different Samplings

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 γ(t i ) = q i 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 α ε (0) = min {3,1 + 2ε} follows [2]. On the other hand, for λ = 1 cumulative chords render a cubic convergence order α(1) = 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 α(λ) = 1 eventuate. Thus no acceleration in α(λ) < α(1) = 3 prevails while λ ∈ [0,1). Finally, another recent result [5] proves that for all λ ∈ [0,1) and ε-uniform samplings, the respective accelerated orders α ε (λ) = min {3,1 + 2ε} are independent of λ. The latter extends the case of α ε (λ = 0) = 1 + 2ε to all λ ∈ [0,1). We revisit here [4] and [5] and verify their sharpness experimentally.