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.
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Gousenbourger, PY., Jacques, L., Absil, PA. (2017). 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 \(\mathbb {S}^2\)?. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2017. Lecture Notes in Computer Science(), vol 10589. Springer, Cham. https://doi.org/10.1007/978-3-319-68445-1_69
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