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The graded group action framework for sub-riemannian orbit models in shape spaces

Thomas Pierron, Alain Trouvé

Abstract

In the standard orbit model on shape analysis, a group of diffeomorphism on the ambient space equipped with a right invariant sub-riemannian metric acts on a space of shapes and induces a sub-riemannian structure on various spaces. An important example is given by the Large Deformation Diffeomorphic Metric Mapping (LDDMM) theory that has been developed initially in the context of medical imaging and image registration. However, the standard theory does not cover many interesting settings emerging in applications. We provide here an extended setting, the graded group action (GGA) framework, specifying regularity conditions to get most of the well known results on the orbit model for general groups and shape spaces equipped with a smooth structure of Banach manifold with application to multi-scale shape spaces. A specific study of the Euler-Poincar{é} equations inside the GCA framework leads to a uniqueness result for the momentum map trajectory lifted from shape spaces with different complexities deciphering possible benefits of over-parametrization in shooting algorithms.

The graded group action framework for sub-riemannian orbit models in shape spaces

Abstract

In the standard orbit model on shape analysis, a group of diffeomorphism on the ambient space equipped with a right invariant sub-riemannian metric acts on a space of shapes and induces a sub-riemannian structure on various spaces. An important example is given by the Large Deformation Diffeomorphic Metric Mapping (LDDMM) theory that has been developed initially in the context of medical imaging and image registration. However, the standard theory does not cover many interesting settings emerging in applications. We provide here an extended setting, the graded group action (GGA) framework, specifying regularity conditions to get most of the well known results on the orbit model for general groups and shape spaces equipped with a smooth structure of Banach manifold with application to multi-scale shape spaces. A specific study of the Euler-Poincar{é} equations inside the GCA framework leads to a uniqueness result for the momentum map trajectory lifted from shape spaces with different complexities deciphering possible benefits of over-parametrization in shooting algorithms.

Paper Structure

This paper contains 23 sections, 30 theorems, 155 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Graded group structures
  3. Admissible graded group structure
  4. Adjoint representation
  5. Absolutely continuous curves and differentiable structure
  6. The example of half-Lie groups
  7. Sub-Riemannian geometry on Banach manifolds induced by right-invariant metrics
  8. Definition of the shape space
  9. Horizontal curves and distance
  10. Sub-Riemannian geodesics and critical points of the energy
  11. Hamiltonian flow and normal geodesics
  12. Inexact matching
  13. Intermezzo: Multiscale shape space
  14. Basic framework
  15. Euler-Poincaré equations
  16. ...and 8 more sections

Key Result

Proposition 2.3

For any $k\geq 1$, $G^k$ is a topological group. Morever, we have

Figures (1)

  • Figure :

Theorems & Definitions (82)

  • Definition 2.1: Admissible graded group structure
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6: Evolution equation in $G^k$
  • proof
  • ...and 72 more