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Geodesic Shooting for Computational Anatomy

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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.

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Authors and Affiliations

  1. Center of Imaging Science & Department of Biomedical Engineering, The John Hopkins University, 301 Clark Hall, Baltimore, MD, 21218, USA

    Michael I. Miller

  2. CMLA, ENS de Cachan, 61 Avenue du Président Wilson, 94235, Cachan CEDEX, France

    Alain Trouvé

  3. Center for Imaging Science & Department of Applied Math and Statistics, The Johns Hopkin, 3245 Clark Hall, Baltimore, MD, 21218, USA

    Laurent Younes

Authors
  1. Michael I. Miller
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  2. Alain Trouvé
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  3. Laurent Younes
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Correspondence to Michael I. Miller.

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Miller, M.I., Trouvé, A. & Younes, L. Geodesic Shooting for Computational Anatomy. J Math Imaging Vis 24, 209–228 (2006). https://doi.org/10.1007/s10851-005-3624-0

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