C1 interpolating Bézier path on Riemannian manifolds, with applications to 3D shape space

摘要

This paper introduces a new framework to fit a C1 Bézier path to a given finite set of ordered data points on shape space of curves. We prove existence and uniqueness properties of the path and give a numerical method for constructing an optimal solution. Furthermore, we present a conceptually simple method to compute the optimal intermediate control points that define this path. The main property of the method is that when the manifold reduces to a Euclidean space or the finite dimensional sphere, the control points minimize the mean square acceleration. Potential applications of fitting smooth paths on Riemannian manifold include applications in robotics, animations, graphics, and medical studies. In this paper, we will focus on different medical applications to predict missing data from few ordered observations.