Journal
JOURNAL OF MATHEMATICAL IMAGING AND VISION
Volume 61, Issue 5, Pages 645-671Publisher
SPRINGER
DOI: 10.1007/s10851-018-0865-2
Keywords
Riemannian manifold; Data fitting; Curve fitting; Interpolation; Smoothing; Blended cubic spline
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Funding
- Fonds de la Recherche Scientifique-FNRS
- Fonds Wetenschappelijk Onderzoek-Vlaanderen under EOS [30468160]
- Communaute francaise de Belgique - Actions de Recherche Concertees [ARC 14/19-060]
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We propose several methods that address the problem of fitting a C1 curve to time-labeled data points on a manifold. The methods have a parameter, , 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 represent 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 (t) at any t requires a small number of exponentials and logarithms (independent of the number of data points). Among the proposed methods, the blended cubic spline technique combines the additional properties of interpolating the data when and reducing to the well-known cubic smoothing spline when the manifold is Euclidean. The methods are illustrated on synthetic and real data.
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