Skip to Main content Skip to Navigation
Conference papers

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(ti) = 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 αε(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 αε(λ) =  min {3,1 + 2ε} are independent of λ. The latter extends the case of αε(λ = 0) = 1 + 2ε to all λ ∈ [0,1). We revisit here [4] and [5] and verify their sharpness experimentally.
Complete list of metadata

Cited literature [15 references]  Display  Hide  Download

https://hal.inria.fr/hal-01496089
Contributor : Hal Ifip <>
Submitted on : Monday, March 27, 2017 - 11:01:56 AM
Last modification on : Thursday, May 2, 2019 - 3:00:17 PM
Long-term archiving on: : Wednesday, June 28, 2017 - 1:02:17 PM

File

978-3-642-40925-7_40_Chapter.p...
Files produced by the author(s)

Licence


Distributed under a Creative Commons Attribution 4.0 International License

Identifiers

Citation

Ryszard Kozera, Lyle Noakes, Piotr Szmielew. Trajectory Estimation for Exponential Parameterization and Different Samplings. 12th International Conference on Information Systems and Industrial Management (CISIM), Sep 2013, Krakow, Poland. pp.430-441, ⟨10.1007/978-3-642-40925-7_40⟩. ⟨hal-01496089⟩

Share

Metrics

Record views

206

Files downloads

219