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

## TL;DR

This paper introduces a discrete Riemannian framework for optimal matching of manifold-valued curves, providing an efficient algorithm that converges to the continuous model and is applicable to various geometries like hyperbolic, Euclidean, and spherical spaces.

## Contribution

It develops a novel discrete Riemannian structure for shape analysis of manifold-valued curves, with proven convergence to the smooth case and practical algorithms for optimal matching.

## Key findings

- The discrete framework accurately approximates the continuous model.
- The algorithm effectively matches curves in hyperbolic, Euclidean, and spherical spaces.
- Comparison with dynamic programming shows competitive performance.

## Abstract

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.