Variational time discretization of Riemannian splines
This work provides a theoretical foundation for spline interpolation on Riemannian manifolds, which is relevant for geometry processing and shape analysis, but the results are largely theoretical and incremental.
The authors generalize cubic splines to Riemannian manifolds by defining spline curves as minimizers of a spline energy, and develop a variational time discretization leading to a constrained optimization problem. They prove existence of continuous and discrete spline curves and establish convergence of discrete to continuous splines via Γ-convergence, with applications in surface processing and shape manifolds.
We investigate a generalization of cubic splines to Riemannian manifolds. Spline curves are defined as minimizers of the spline energy - a combination of the Riemannian path energy and the time integral of the squared covariant derivative of the path velocity - under suitable interpolation conditions. A variational time discretization for the spline energy leads to a constrained optimization problem over discrete paths on the manifold. Existence of continuous and discrete spline curves is established using the direct method in the calculus of variations. Furthermore, the convergence of discrete spline paths to a continuous spline curve follows from the $Γ$-convergence of the discrete to the continuous spline energy. Finally, selected example settings are discussed, including splines on embedded finite-dimensional manifolds, on a high-dimensional manifold of discrete shells with applications in surface processing, and on the infinite-dimensional shape manifold of viscous rods.