A variational model for data fitting on manifolds by minimizing the acceleration of a Bézier curve
This work provides a novel method for curve fitting on manifolds, benefiting applications in computer graphics, robotics, and geometric data analysis.
The authors derive a variational model for fitting composite Bézier curves to data on Riemannian manifolds, minimizing mean squared acceleration while staying close to data points. Their algorithm outperforms previous methods in both interpolation and approximation tasks.
We derive a variational model to fit a composite Bézier curve to a set of data points on a Riemannian manifold. 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, expressed as a concatenation of so-called adjoint Jacobi fields. Several examples illustrate the capabilites and validity of this approach both for interpolation and approximation. The examples also illustrate that the approach outperforms previous works tackling this problem.