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A Log-Euclidean Framework for Statistics on Diffeomorphisms

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Abstract

In this article, we focus on the computation of statistics of invertible geometrical deformations (i.e., diffeomorphisms), based on the generalization to this type of data of the notion ofprincipal logarithm. Remarkably, this logarithm is a simple 3D vector field, and is well-defined for diffeomorphisms close enough to the identity. This allows to performvectorial statistics on diffeomorphisms, while preserving the invertibility constraint, contrary to Euclidean statistics on displacement fields. We also present here two efficient algorithms to compute logarithms of diffeomorphisms and exponentials of vector fields, whose accuracy is studied on synthetic data. Finally, we apply these tools to compute the mean of a set of diffeomorphisms, in the context of a registration experiment between an atlas an a database of 9 T1 MR images of the human brain.

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Author information

Authors and Affiliations

  1. INRIA Sophia – Epidaure Project, 2004 Route des Lucioles, BP 93, 06902 Cedex, Sophia Antipolis, France

    Vincent Arsigny, Olivier Commowick, Xavier Pennec & Nicholas Ayache

  2. DOSISoft S.A., 45 Avenue Carnot, 94 230, Cachan, France

    Olivier Commowick

Authors
  1. Vincent Arsigny

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  2. Olivier Commowick

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  3. Xavier Pennec

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  4. Nicholas Ayache

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Editor information

Editors and Affiliations

  1. Department of Informatics and Mathematical Modelling, Technical University of Denmark, Denmark

    Rasmus Larsen

  2. Nordic Bioscience, Herlev, Denmark

    Mads Nielsen

  3. Department of Computer Science, University of Copenhagen, Denmark

    Jon Sporring

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© 2006 Springer-Verlag Berlin Heidelberg

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Arsigny, V., Commowick, O., Pennec, X., Ayache, N. (2006). A Log-Euclidean Framework for Statistics on Diffeomorphisms. In: Larsen, R., Nielsen, M., Sporring, J. (eds) Medical Image Computing and Computer-Assisted Intervention – MICCAI 2006. MICCAI 2006. Lecture Notes in Computer Science, vol 4190. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11866565_113

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