Quartic orders and sharpness in trajectory estimation for smooth cumulative chord cubics

Abstract

This paper discusses the issue of fitting reduced data with smooth interpolant by piecewise-cubics to estimate an unknown curve γ in arbitrary Euclidean space. The interpolation knots satisfying γ(ti) = qi are assumed to be unknown and guessed according to so-called cumulative chords. More specifically, first estimates of the derivatives at interpolation points Qm are found by using piecewise-cubics combined with cumulative chords. Next Hermite interpolation is applied to construct C1 piecewise-cubic yielding the interpolant with no cusps and corners. At least quartic orders of convergence for both trajectory and length estimations are analytically established in [1]. However their sharpness was exclusively tested for length estimation. This paper verifies experimentally the latter with respect to the trajectory approximation. Additionally, an excellent performance of the interpolant in question on sparse data is also confirmed experimentally. Fitting reduced data is used in computer vision and graphics, engineering, physics as well as in medical and biological sciences. A specific example of medical application is also presented in this paper.