Sharpness in Trajectory Estimation by Piecewise-quadratics(-cubics) and Cumulative Chords
In this paper we verify sharpness of the theoretical results concerning the asymptotic orders of trajectory approximation of the unknown parametric curve γ in arbitrary Euclidean space. The pertinent interpolation schemes (based on piecewise-quadratics and piecewise-cubics) are here considered for the so-called reduced data. The latter forms an ordered collection of points without provision of the associated interpolation knots. To complement such data i.e. to determine the missing knots, cumulative chord parameterization is invoked. Sharpness of cubic and quartic orders of convergence are demonstrated for piecewise-quadratics and piecewise-cubics, respectively. This topic has its ramification in computer vision (e.g. image segmentation), computer graphics (e.g. trajectory modeling) or in engineering (e.g. robotics).
KeywordsConvergence Rate Interpolation Scheme Length Estimation Fast Convergence Rate Asymptotic Order
Unable to display preview. Download preview PDF.
- 5.Kozera, R.: Curve modeling via interpolation based on multidimensional reduced data. Studia Informatica 25(4B(61)), 1–140 (2004)Google Scholar
- 6.Janik, M., Kozera, R., Kozioł, P.: Reduced data for curve modeling - applications in graphics. Computer Vision and Physics (submitted)Google Scholar
- 7.Noakes, L., Kozera, R.: Cumulative chords and piecewise-quadratics and piecewise-cubics. In: Klette, R., Kozera, R., Noakes, L., Weickert, J. (eds.) Geometric Properties of Incomplete Data. Computational Imaging and Vision, The Netherlands, vol. 31, pp. 59–75 (2006)Google Scholar
- 9.Kvasov, B.: Methods of Shape-Preserving Spline Approximation. World Scientific, Singapore (2000)Google Scholar