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Optimal matching between curves in a manifold

Alice Le Brigant, Marc Arnaudon, Frédéric Barbaresco

TL;DR

A simple algorithm allowing to compute geodesics of the quotient shape space using a canonical decomposition of a path in the associated principal bundle is introduced.

Abstract

This paper is concerned with the computation of an optimal matching between two manifold-valued curves. Curves are seen as elements of an infinite-dimensional manifold and compared using a Riemannian metric that is invariant under the action of the reparameterization group. This group induces a quotient structure classically interpreted as the ''shape space''. We introduce a simple algorithm allowing to compute geodesics of the quotient shape space using a canonical decomposition of a path in the associated principal bundle. We consider the particular case of elastic metrics and show simulations for open curves in the plane, the hyperbolic plane and the sphere.

Optimal matching between curves in a manifold

TL;DR

A simple algorithm allowing to compute geodesics of the quotient shape space using a canonical decomposition of a path in the associated principal bundle is introduced.

Abstract

This paper is concerned with the computation of an optimal matching between two manifold-valued curves. Curves are seen as elements of an infinite-dimensional manifold and compared using a Riemannian metric that is invariant under the action of the reparameterization group. This group induces a quotient structure classically interpreted as the ''shape space''. We introduce a simple algorithm allowing to compute geodesics of the quotient shape space using a canonical decomposition of a path in the associated principal bundle. We consider the particular case of elastic metrics and show simulations for open curves in the plane, the hyperbolic plane and the sphere.
Paper Structure (4 sections, 4 theorems, 22 equations, 5 figures, 2 algorithms)

This paper contains 4 sections, 4 theorems, 22 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. The optimal matching algorithm
  3. Example : elastic metrics
  4. Simulations

Key Result

Proposition 1

The horizontal part of a path of curves $c$ is at most the same length as $c$

Figures (5)

  • Figure 1: Schematic representation of the shape bundle.
  • Figure 2: Geodesics between parameterized curves (blue) and corresponding horizontal geodesics (red) in the hyperbolic half-plane, and their superpositions.
  • Figure 3: Length of the geodesics of the hyperbolic half-plane shown in Figure \ref{['fig:horgeodshootH2']}.
  • Figure 4: Initial and horizontal geodesics between spherical parameterized curves.
  • Figure 5: Superposition of the initial (blue) and horizontal (red) geodesics obtained for different sets of parameterizations of three pairs of plane curves.

Theorems & Definitions (8)

  • Proposition 1
  • proof
  • Proposition 2: Horizontal part of a path
  • proof
  • Proposition 3: Horizontal part of a vector for an elastic metric
  • proof
  • Proposition 4: Horizontal part of a path for an elastic metric
  • proof