On Restricting Planar Curve Evolution to Finite Dimensional Implicit Subspaces with Non-Euclidean Metric

  • Authors:

  • Aditya Tatu;François Lauze;Stefan Sommer;Mads Nielsen

  • Affiliations:

  • Institute of Computer Science, University of Copenhagen, Copenhagen, Denmark;Institute of Computer Science, University of Copenhagen, Copenhagen, Denmark;Institute of Computer Science, University of Copenhagen, Copenhagen, Denmark;Institute of Computer Science, University of Copenhagen, Copenhagen, Denmark

  • Venue:
  • Journal of Mathematical Imaging and Vision
  • Year:
  • 2010

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Abstract

This paper deals with restricting curve evolution to a finite and not necessarily flat space of curves, obtained as a subspace of the infinite dimensional space of planar curves endowed with the usual but weak parametrization invariant curve L 2-metric.We first show how to solve differential equations on a finite dimensional Riemannian manifold defined implicitly as a submanifold of a parameterized one, which in turn may be a Riemannian submanifold of an infinite dimensional one, using some optimal control techniques.We give an elementary example of the technique on a spherical submanifold of a 3-sphere and then a series of examples on a highly non-linear subspace of the space of closed spline curves, where we have restricted mean curvature motion, Geodesic Active contours and compute geodesic between two curves.