Geodesic Shooting for Computational Anatomy

  • Authors:

  • Michael I. Miller;Alain Trouvé;Laurent Younes

  • Affiliations:

  • Center of Imaging Science & Department of Biomedical Engineering, The John Hopkins University, Baltimore, USA 21218;CMLA, ENS de Cachan, Cachan CEDEX, France 94235;Center for Imaging Science & Department of Applied Math and Statistics, The Johns Hopkin, Baltimore, USA 21218

  • Venue:
  • Journal of Mathematical Imaging and Vision
  • Year:
  • 2006

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Abstract

Studying large deformations with a Riemannian approach has been an efficient point of view to generate metrics between deformable objects, and to provide accurate, non ambiguous and smooth matchings between images. In this paper, we study the geodesics of such large deformation diffeomorphisms, and more precisely, introduce a fundamental property that they satisfy, namely the conservation of momentum. This property allows us to generate and store complex deformations with the help of one initial "momentum" which serves as the initial state of a differential equation in the group of diffeomorphisms. Moreover, it is shown that this momentum can be also used for describing a deformation of given visual structures, like points, contours or images, and that, it has the same dimension as the described object, as a consequence of the normal momentum constraint we introduce.