  The aim of this paper is to find an optimal matching between manifold-valued
curves, and thereby adequately compare their shapes, seen as equivalent classes
with respect to the action of reparameterization. Using a canonical
decomposition of a path in a principal bundle, we introduce a simple algorithm
that finds an optimal matching between two curves by computing the geodesic of
the infinite-dimensional manifold of curves that is at all time horizontal to
the fibers of the shape bundle. We focus on the elastic metric studied in the
so-called square root velocity framework. The quotient structure of the shape
bundle is examined, and in particular horizontality with respect to the fibers.
These results are more generally given for any elastic metric. We then
introduce a comprehensive discrete framework which correctly approximates the
smooth setting when the base manifold has constant sectional curvature. It is
itself a Riemannian structure on the product manifold of "discrete curves"
given by a finite number of points, and we show its convergence to the
continuous model as the size of the discretization goes to infinity.
Illustrations of optimal matching between discrete curves are given in the
hyperbolic plane, the plane and the sphere, for synthetic and real data, and
comparison with dynamic programming is established.
