Research Seminar

Curve Modeling via Interpolation Based on Multidimensional Reduced Data Part 1

Abstract

         In both  talks we consider the problem of modeling curves together with estimation of their length and
trajectory via various interpolation schemes (i.e. piecewise polynomials) based on the so-called reduced data (an ordered sequence of interpolation points stripped from the corresponding knot parameters). Such interpolation is termed as non-parametric, as opposed to the classical parametric one, where the interpolation points together with the respective knots are given (non-reduced data). The analysis in question applies to  the general class of smooth curves in arbitrary Euclidean space  sampled with sparse or dense data.

PART A: we compare both parametric and non-parametric interpolations with various guesses of the missing knots based on different choices of interpolation schemes. The corresponding asymptotic rates of convergence to estimate length and trajectory of the unknown curve are established and  shown to match the case of parametric interpolation (for data dense). The illustrative examples and experiments complement the presentation and confirm the sharpness of the derived theoretical results.