Journal
FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
Volume 12, Issue 1, Pages 49-73Publisher
SPRINGER
DOI: 10.1007/s10208-011-9091-7
Keywords
Curve fitting; Steepest-descent; Sobolev space; Palais metric; Geodesic distance; Energy minimization; Splines; Piecewise geodesic; Smoothing; Riemannian center of mass
Funding
- Interuniversity Attraction Poles Programme
- AFOSR [FA9550-06-1-0324]
- ONR [N00014-09-10664]
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [0915003] Funding Source: National Science Foundation
Ask authors/readers for more resources
Given data points p (0),aEuro broken vertical bar,p (N) on a closed submanifold M of ae (n) and time instants 0=t (0)< t (1)< a <...a <...a <...< t (N) =1, we consider the problem of finding a curve gamma on M 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 M. The steepest-descent direction, defined in the sense of the first-order and second-order Palais metric, respectively, is shown to admit analytical expressions involving parallel transport and covariant integral along curves. Illustrations are given in ae (n) and on the unit sphere.
Authors
I am an author on this paper
Click your name to claim this paper and add it to your profile.
Reviews
Recommended
No Data Available