Statistics on the manifold of multivariate normal distributions: Theory and application to diffusion tensor MRI processing

Christophe Lenglet, Mikaël Rousson, Rachid Deriche, Olivier Faugeras

Research output: Contribution to journalArticle

176 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)423-444
Number of pages22
JournalJournal of Mathematical Imaging and Vision
Volume25
Issue number3
DOIs
StatePublished - Oct 1 2006
Externally publishedYes

Keywords

  • Covariance matrix
  • Curvature
  • Fisher information matrix
  • Geodesic distance
  • Geodesics
  • Information geometry
  • Mean
  • Multivariate normal distribution
  • Ricci tensor
  • Riemannian geometry
  • Statistics
  • Symmetric positive-definite matrix

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