by Kozera, R, Noakes, L and Klette, R
Abstract:
This paper studies differences in estimating length (and also trajectory) of an unknown parametric curve γ : [0, 1] → ℝn from an ordered collection of data points qi = γ(ti), with either the ti’s known or unknown. For the ti’s uniform (known or unknown) piecewise Lagrange interpolation provides efficient length estimates, but in other cases it may fail. In this paper, we apply this classical algorithm when the ti’s are sampled according to first α-order and then when sampling is ε-uniform. The latter was introduced in [20] for the case where the ti’s are unknown. In the present paper we establish new results for the case when the ti’s are known for both types of samplings. For curves sampled ε-uniformly, comparison is also made between the cases, where the tabular parameters ti’s are known and unknown. Numerical experiments are carried out to investigate sharpness of our theoretical results. The work may be of interest in computer vision and graphics, approximation and complexity theory, digital and computational geometry, and digital image analysis. © Springer-Verlag Berlin Heidelberg 2003.
Reference:
External versus internal parameterizations for lengths of curves with nonuniform samplings (Kozera, R, Noakes, L and Klette, R), volume 2616, 2003.
Bibtex Entry:
@book{kozera2003externalsamplings,
author = "Kozera, R and Noakes, L and Klette, R",
journal = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",
pages = "403--418",
publisher = "",
title = "External versus internal parameterizations for lengths of curves with nonuniform samplings",
volume = "2616",
year = "2003",
abstract = "This paper studies differences in estimating length (and also trajectory) of an unknown parametric curve γ : [0, 1] → ℝn from an ordered collection of data points qi = γ(ti), with either the ti's known or unknown. For the ti's uniform (known or unknown) piecewise Lagrange interpolation provides efficient length estimates, but in other cases it may fail. In this paper, we apply this classical algorithm when the ti's are sampled according to first α-order and then when sampling is ε-uniform. The latter was introduced in [20] for the case where the ti's are unknown. In the present paper we establish new results for the case when the ti's are known for both types of samplings. For curves sampled ε-uniformly, comparison is also made between the cases, where the tabular parameters ti's are known and unknown. Numerical experiments are carried out to investigate sharpness of our theoretical results. The work may be of interest in computer vision and graphics, approximation and complexity theory, digital and computational geometry, and digital image analysis. © Springer-Verlag Berlin Heidelberg 2003.",
issn = "0302-9743",
eissn = "1611-3349",
language = "eng",
}