Journal of Mathematical Imaging and Vision

, Volume 25, Issue 3, pp 423–444 | Cite as

Statistics on the Manifold of Multivariate Normal Distributions: Theory and Application to Diffusion Tensor MRI Processing

  • Christophe Lenglet
  • Mikaël Rousson
  • Rachid Deriche
  • Olivier Faugeras


This paper is dedicated to the statistical analysis of the space of multivariate normal distributions with an application to the processing of Diffusion Tensor Images (DTI). It relies on the differential geometrical properties of the underlying parameters space, endowed with a Riemannian metric, as well as on recent works that led to the generalization of the normal law on Riemannian manifolds. We review the geometrical properties of the space of multivariate normal distributions with zero mean vector and focus on an original characterization of the mean, covariance matrix and generalized normal law on that manifold. We extensively address the derivation of accurate and efficient numerical schemes to estimate these statistical parameters. A major application of the present work is related to the analysis and processing of DTI datasets and we show promising results on synthetic and real examples.


multivariate normal distribution symmetric positive-definite matrix information geometry Riemannian geometry Fisher information matrix geodesics geodesic distance Ricci tensor curvature statistics mean covariance matrix diffusion tensor magnetic resonance imaging 


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Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  • Christophe Lenglet
    • 1
  • Mikaël Rousson
    • 2
  • Rachid Deriche
    • 1
  • Olivier Faugeras
    • 1
  1. 1.I.N.R.I.A Sophia-AntipolisSophia-AntipolisFrance
  2. 2.Siemens Corporate ResearchPrincetonUSA

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