Length Estimation for Exponential Parameterization and ε-Uniform Samplings

• Ryszard Kozera
• Lyle Noakes
• Piotr Szmielew
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8334)

Abstract

This paper discusses the problem of estimating the length of the unknown curve γ in Euclidean space, from ε-uniformly (for ε ≥ 0) sampled reduced data $$Q_m=\{q_i\}_{i=0}^m$$, where γ(t i ) = q i . The interpolation knots $$\{t_i\}_{i=0}^m$$ are assumed here to be unknown (yielding the so-called non-parametric interpolation). We fit Q m with the piecewise-quadratic interpolant $$\hat \gamma_2$$ combined with the so-called exponential parameterization (characterized by the parameter λ ∈ [0,1]). Such parameterization (applied e.g. in computer graphics for curve modeling [1], [2]) uses estimates of the missing knots $$\{t_i\}_{i=0}^m\approx\{\hat t_i\}_{i=0}^m$$. The asymptotic orders β ε (λ) for length estimation $$d(\gamma)\approx d(\hat \gamma_2)$$ in case of λ = 0 (uniformly guessed knots) read as β ε (0) =  min {4,4ε} (for ε > 0) - see [3]. On the other hand λ = 1 (cumulative chords) renders β ε (1) =  min {4,3 + ε} (see [4]). A recent result [5] proves that for all λ ∈ [0,1) and ε-uniform samplings, the respective orders amount to β ε (λ) =  min {4,4ε}. As such β ε (λ) are independent of λ ∈ [0,1). In addition, the latter renders a discontinuity in asymptotic orders β ε (λ) at λ = 1. In this paper we verify experimentally the above mentioned theoretical results established in [5].

Keywords

Length estimation interpolation numerical analysis computer graphics and vision

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