Paper ID sheet UCL-INMA-2018.04


Data fitting on manifolds with composite Bézier-like curves and blended cubic splines

Pierre-Yves Gousenbourger, Estelle Massart, P.-A. Absil
We propose several methods that address the problem of fitting a $C^1$ curve $\gamma$ to time-labeled data points on a manifold. The methods have a parameter, $\lambda$, to adjust the relative importance of the two goals that the curve should meet: being "straight enough" while fitting the data "closely enough". The methods are designed for ease of use: they only require to compute Riemannian exponentials and logarithms, they repre- sent the curve by means of a number of tangent vectors that grows linearly with the number of data points, and, once the representation is computed, evaluating $\gamma(t)$ at any $t$ requires a small number of exponentials and log- arithms (independent of the number of data points). Among the proposed methods, the blended cubic spline technique combines the additional properties of inter- polating the data when $\lambda \to \infty$ and reducing to the well-known cubic smoothing spline when the manifold is Euclidean. The methods are illustrated on synthetic and real data.
Key words
Riemannian manifold; data fitting; curve fitting; interpolation; smoothing; blended cubic spline
Journal of Mathematical Imaging and Vision volume 61, pages 645–671 (2019)