Paper ID sheet
- TITLE: A gradient-descent method for curve fitting on Riemannian manifolds
- AUTHORS: Chafik Samir, P.-A. Absil, Anuj Srivastava, Eric Klassen
- ABSTRACT:
Given data points $p_0,\ldots,p_N$ on a manifold $\calM$ and time
instants $0=t_0<t_1<\ldots<t_N=1$, we consider the problem of
finding a curve $\gamma$ on $\calM$ that best approximates the data
points at the given instants while being as ``regular'' as
possible. Specifically, $\gamma$ is expressed as the curve that
minimizes the weighted sum of a sum-of-squares term penalizing the
lack of fitting to the data points and a regularity term defined, in
the first case as the mean squared velocity of the curve, and in the
second case as the mean squared acceleration of the curve. In both
cases, the optimization task is carried out by means of a
steepest-descent algorithm on a set of curves on $\calM$. The
steepest-descent direction, defined in the sense of the first-order
and second-order Palais metric, respectively, is shown to admit
simple formulas.
- KEY WORDS: curve fitting, steepest-descent, Sobolev space,
Palais metric, geodesic distance, energy minimization, splines,
piecewise geodesic, smoothing, Karcher mean
- STATUS: Technical report UCL-INMA-2009.023, registered 28 August
2009. Submitted for publication on 28 August 2009.
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