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Journal of Mathematical Imaging and Vision

, Volume 61, Issue 1, pp 40–70 | Cite as

A Discrete Framework to Find the Optimal Matching Between Manifold-Valued Curves

  • Alice Le BrigantEmail author
Article

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 Le Brigant (J Geom Mech 9(2):131–156, 2017) using 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 \(M^{n}\) of “discrete curves” given by n points, and we show its convergence to the continuous model as the size n of the discretization goes to \(\infty \). Illustrations of geodesics and optimal matchings between discrete curves are given in the hyperbolic plane, the plane and the sphere, for synthetic and real data, and comparison with dynamic programming (Srivastava and Klassen in Functional and shape data analysis, Springer, Berlin, 2016) is established.

Keywords

Shape analysis Optimal matching Manifold-valued curves Discretization 

Notes

Acknowledgements

This research was supported by Thales Air Systems and the french MoD DGA. We also acknowledge the strong support of the European Commission, Airbus and the Airlines (Lufthansa, Air-France, Austrian, Air Namibia, Cathay Pacic, Iberia and China Airlines so far) who carry the MOZAIC or IAGOS equipment and perform the maintenance since 1994. MOZAIC is presently funded by INSU-CNRS (France), Météo-France, Université Paul Sabatier (Toulouse, France) and Research Center Jülich (FZJ, Jülich, Germany). IAGOS has been and is additionally funded by the EU projects IAGOS-DS and IAGOS-ERI. The MOZAIC-IAGOS database is supported by ETHER (CNES and INSU-CNRS). Data are also available via Ether web site http://www.pole-ether.fr.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut de Mathématiques de Bordeaux, UMR 5251, CNRSUniversité de BordeauxBordeauxFrance

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