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

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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.

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Authors and Affiliations

  1. I.N.R.I.A Sophia-Antipolis, 2004 Route des lucioles, Sophia-Antipolis, France

    Christophe Lenglet, Rachid Deriche & Olivier Faugeras

  2. Siemens Corporate Research, 755 College Road East, Princeton, NJ, 08540, USA

    Mikaël Rousson

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  1. Christophe Lenglet
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  4. Olivier Faugeras
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Correspondence to Christophe Lenglet.

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Christophe Lenglet is a Ph.D. candidate in biomedical imaging at INRIA Sophia-Antipolis. He received his M.S. in applied mathematics and computer vision from Ecole Normale Supérieure de Cachan in 2003. Prior to that, he studied computer science at Compiègne University of Technology from where he graduated in 2003 with a minor in cognitive science. His research interests include MR neuro-imaging, brain connectivity mapping, segmentation and registration techniques, differential geometry and variational methods. He serves as a reviewer for international journals and conferences in image processing/computer vision and medical imaging.

Mikaël Rousson was born in Annonay, France, in 1978. He graduated from Ecole Supérieure en Sciences Informatiques, Sophia-Antipolis, France in 2001. In 2004, he received the PhD degree in computer science and signal processing from the University of Nice, Sophia-Antipolis, France. Since November 2004, he has been working as a research scientist at Siemens Corporate Research, Princeton, NJ. His research interests include geometric methods, shape and image statistics.

Rachid Deriche graduated from Ecole Nationale Supérieure des Télécommunications, Paris, in 1979 and received the Ph.D degree in Mathematics from the University of Paris IX, Dauphine in 1982. He is currently a Research Director at INRIA Sophia-Antipolis in the Computer and Biological Vision Group Odyssée. His research interests are in Image Processing, Computer and Biological Vision and include in particular the area related to variational methods and partial differential equations for vision. More generally, he is very interested by the application of mathematics to Image Processing, Computer and Biological Vision. He has authored and co-authored more than 120 scientific papers.

Olivier Faugeras is a graduate from the Ecole Polytechnique (1971). He holds a PhD in Computer Science and Electrical Engineering from the University of Utah (1976) and a Doctorate of Science from Paris VI University (1981). He is currently Research Director at INRIA (National Research Institute in Computer Science and Control Theory), where he leads the Odyssée laboratory located in Sophia-Antipolis and Ecole Normale Supérieure, Paris. His research interests include the application of mathematics to computer and biological vision, shape representation and recognition, the use of functional imaging (MR, MEG, EEG) for understanding brain activity and in particular visual perception. He has published extensively in archival Journals, International Conferences, has contributed chapters to many books and is the author of “Artificial 3-D Vision” published in 1993 by MIT Press and, with Quang-Tuan Luong and Theo Papadopoulo, of “The Geometry of Multiple Images” which appeared in March 2001, also at MIT Press. He was an adjunct Professor from 1996 to 2001 in the Electrical Engineering and Computer Science Department of the Massachusetts Institute of Technology and a member of the AI Lab. He is an Associate Editor of several international scientific Journals including Machine Vision and Applications, Videre, Image and Vision Computing. He has served as Associate Editor for IEEE PAMI from 1987 to 1990 and as co-Editor-in-Chief of the International Journal of Computer Vision from 1991 to 2004. In April 1989 he received the “Institut de France - Fondation Fiat” award from the french Academy of Sciences for his work in Vision and Robotics. In July 1998 he received the “France Telecom” award from the french Academy of Sciences for his work on Computer Vision and Geometry. In November 1998 he was elected a member of the french Academy of Sciences.

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Lenglet, C., Rousson, M., Deriche, R. et al. Statistics on the Manifold of Multivariate Normal Distributions: Theory and Application to Diffusion Tensor MRI Processing. J Math Imaging Vis 25, 423–444 (2006). https://doi.org/10.1007/s10851-006-6897-z

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