A Modified Complete Spline Interpolation and Exponential Parameterization

Abstract

In this paper a modified complete spline interpolation based on reduced data is examined in the context of trajectory approximation. Reduced data constitute an ordered collection of interpolation points in arbitrary Euclidean space, stripped from the corresponding interpolation knots. The exponential parameterization (controlled by $\lambda \in [0,1]$) compensates the above loss of information and provides specific scheme to approximate the distribution of the missing knots. This approach is commonly used in computer graphics or computer vision in curve modeling and image segmentation or in biometrics for feature extraction. The numerical verification of asymptotic orders $\alpha (\lambda )$ in trajectory estimation by modified complete spline interpolation is performed here for regular curves sampled more-or-less uniformly with the missing knots parameterized according to exponential parameterization. Our approach is equally applicable to either sparse or dense data. The numerical experiments confirm a slow linear convergence orders $\alpha (\lambda )=1$ holding for all $\lambda \in [0,1)$ and a quartic one $\alpha (1)=4$ once modified complete spline is used. The paper closes with an example of medical image segmentation.