Fitting Curves on Riemannian Manifolds Using Energy Minimization
نویسندگان
چکیده
Given data points p0, . . . , pN on a Riemannian manifold M and time instants 0 = t0 < t1 < . . . < tN = 1, we consider the problem of finding the curve γ on M that best approximates the data points at the given instants. In this work, γ is expressed as the curve that minimizes the weighted sum of a least-squares term penalizing the lack of fitting to the data points and a regularity term defined as the mean squared velocity of the curve. The optimization task is carried out by means of a steepest-descent algorithm on the set of continuous paths on M. The steepest-descent direction, defined in the sense of the Palais metric, is shown to admit a simple formula based on parallel translation.
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