Kalender Department Mathematik

Tue, 11.12.2018, 12:00
A variational model for data fitting on manifolds by minimizing the acceleration of a Bézier curve

Referent: PD Dr. Ronny Bergmann, TU Chemnitz
Veranstalter: Martin Burger
Raum: Hörsaal H12

Fitting a smooth curve to data points $d_0,\dots,d_n$ lying on a Riemannian manifold $\mathcal M$ and associated with real-valued parameters $t_0,\dots,t_n$ is a common problem in applications like wind field approximation, rigid body motion interpolation, or sphere-valued data analysis. The resulting curve should strike a balance between data proximity and a smoothing regularization constraint.

In this talk we present the general framework of optimization on manifolds. We then introduce a variational model to fit a composite Bézier curve to the set of data points $d_0,\dots,d_n$ on a Riemannian manifold $\mathcal M$. The resulting curve is obtained in such a way that its mean squared acceleration is minimal in addition to remaining close the data points. We approximate the acceleration by discretizing the squared second order derivative along the curve. We derive a closed-form, numerically stable and efficient algorithm to compute the gradient of a Bézier curve on manifolds with respect to its control points. This gradient can be expressed as a concatenation of so called adjoint Jacobi fields. Several examples illustrate the capabilities of this approach both for interpolation and approximation.