A variational model for data fitting on manifolds by minimizing the acceleration of a Bézier curve

Abstract

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.

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• Sasaki Metric for Spline Models of Manifold-Valued Trajectories
• Fenchel Duality Theory and A Primal-Dual Algorithm on Riemannian Manifolds
• Intrinsic shape analysis in archaeology: A case study on ancient sundials
• First- and Second-Order Analysis for Optimization Problems with Manifold-Valued Constraints
• Approximating maps into manifolds with lower curvature bounds
• Bi-invariant Geodesic Regression with Data from the Osteoarthritis Initiative