A discrete framework to find the optimal matching between manifold-valued curves

This paper introduces a detailed discrete framework to study curves on a manifold of constant sectional curvature regardless of parameterization. This model results from the discretization of the elastic metric G^{1,1/2} studied in the square root velocity framework extended to smooth manifold-valued curves, and is itself a Riemannian structure on the product manifold M^{n+1} of "discrete curves" given by n + 1 points. We show that the discrete energy of a discretization of size n of a path of smooth curves converges to the continuous energy as n $\rightarrow$ $\infty$. We also study the quotient structure of the space of unparameterized curves (or shapes) of the continuous model, and characterize the associated horizontal subspace of the tangent bundle. We introduce a simple algorithm that constructs the horizontal geodesic between two parameterized curves using a canonical decomposition of a path in a principal bundle. Illustrations are given for curves in the hyperbolic plane M = H^2 , the plane M = R^2 and the sphere M = S^2 .