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

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