Paper ID sheet UCLINMA2018.04
 Title

Data fitting on manifolds with composite Bézierlike curves and blended cubic splines
 Authors
 PierreYves Gousenbourger, Estelle Massart, P.A. Absil
 Abstract

We propose several methods that address
the problem of fitting a $C^1$ curve $\gamma$ to timelabeled 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
wellknown 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
 Status
 Journal of Mathematical Imaging and Vision volume 61, pages 645–671 (2019)
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