Geometric optimization methods for adaptive filtering
Geometric optimization methods for adaptive filtering
Curves and surfaces for CAGD: a practical guide
Curves and surfaces for CAGD: a practical guide
The De Casteljau Algorithm on Lie Groups and Spheres
Journal of Dynamical and Control Systems
International Journal of Systems Science - The Seventh Portuguese Conference on Automatic Control (Controlo'2006)
International Journal of Systems Science - The Seventh Portuguese Conference on Automatic Control (Controlo'2006)
Higher-order smoothing splines versus least squares problems on Riemannian manifolds
Journal of Dynamical and Control Systems
Fitting smoothing splines to time-indexed, noisy points on nonlinear manifolds
Image and Vision Computing
An intrinsic formulation of the problem on rolling manifolds
Journal of Dynamical and Control Systems
Averaging complex subspaces via a Karcher mean approach
Signal Processing
Hi-index | 0.00 |
We present a procedure to generate smooth interpolating curves on submanifolds, which are given in closed form in terms of the coordinates of the embedding space. In contrast to other existing methods, this approach makes the corresponding algorithm easy to implement. The idea is to project the prescribed data on the manifold onto the affine tangent space at a particular point, solve the interpolation problem on this affine subspace, and then project the resulting curve back on the manifold. One of the novelties of this approach is the use of rolling mappings. The manifold is required to roll on the affine subspace like a rigid body, so that the motion is described by the action of the Euclidean group on the embedding space. The interpolation problem requires a combination of a pullback/push forward with rolling and unrolling. The rolling procedure by itself highlights interesting properties and gives rise to a new, but simple, concept of geometric polynomial curves on manifolds. This paper is an extension of our previous work, where mainly the 2-sphere case was studied in detail. The present paper includes results for the n-sphere, orthogonal group SO n , and real Grassmann manifolds. In particular, we present the kinematic equations for rolling these manifolds along curves without slip or twist, and derive from them formulas for the parallel transport of vectors along curves on the manifold.