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.
Bibliographical noteFunding Information:
The authors would like to thank G. Sapiro (Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, USA), S. Lehéricy and K. Ugurbil (Center for Magnetic Resonance Research, University of Minnesota, Minneapolis, USA) for their valuable comments and their expertise to acquire the data used in this paper. This research was partially supported by grants NSF-0404617 US-France (INRIA) Cooperative Research, NIH-R21-RR019771, NIH-RR08079, the MIND Institute, the Keck foundation and the Région Provence-Alpes-Côte d’Azur.
- Covariance matrix
- Fisher information matrix
- Geodesic distance
- Information geometry
- Multivariate normal distribution
- Ricci tensor
- Riemannian geometry
- Symmetric positive-definite matrix