Fast Method to Fit a $\mathcal {C}^1$ Piecewise-Bézier Function to Manifold-Valued Data Points: How Suboptimal is the Curve Obtained on the Sphere $\mathbb {S}^2$?

Abstract

We propose an analysis of the quality of the fitting method proposed in [7]. This method fits smooth paths to manifold-valued data points using $\mathcal {C}^1$ piecewise-Bézier functions. This method is based on the principle of minimizing an objective function composed of a data-attachment term and a regularization term chosen as the mean squared acceleration of the path. However, the method strikes a tradeoff between speed and accuracy by following a strategy that is guaranteed to yield the optimal curve only when the manifold is linear. In this paper, we focus on the sphere $\mathbb {S}^2$. We compare the quality of the path returned by the algorithms from [7] with the path obtained by minimizing, over the same search space of $\mathcal {C}^1$ piecewise-Bézier curves, a finite-difference approximation of the objective function by means of a derivative-free manifold-based optimization method.