Seminars/Colloquia/Invited Talks

Seminars

Xavier Pennec

Statistical Computing on Manifolds: From Riemannian Geometry to Computational Anatomy

PLACE:	Clark 314
EVENT:	CIS Seminar Series
DATE:	April 11, 2006
TIME:	1:00 - 2:00

Abstract

Measurements of geometric primitives are often noisy in real applications and we need to use statistics either to reduce the uncertainty (estimation), to compare measurements, or to test hypotheses. Unfortunately, geometric primitives often belong to manifolds that are not vector spaces. Based on a Riemannian manifold structure, we develop the notions of mean value and covariance matrix of a random element, normal law, Mahalanobis distance and test. We also provide an efficient algorithm to compute the mean value, and tractable approximations (with their limits) of the generalized normal law for small variances. This show that we can effectively implement and work with these definitions. We exemplify first this framework with the evaluation of the performances of 3D rigid registration algorithms (i.e. statistics on rigid-body transformations). To establish a reference (a Bronze Standard) in the absence of a Gold Standard, we propose a multiple registration protocol involving as many methods as possible so that the (possible) bias of each method becomes a random variable and is taken into account in the final mean registration. Results show that we are able to show subpixel accuracy and to detect very small biases, such as a bias in translation a 10th of a voxel due to the chemical shift, or a tilt of 2 degree of the images.
Then, we extend the Riemannian computing framework to PDEs forsmoothing and interpolation of fields of features with the example of positive define symmetric matrices (tensors). These covariance matrices are used to in Diffusion Tensor Imaging or to describe the joint anatomical variability at different places (Green function) in shape variability analysis. As symmetric positive definite matrices constitute a convex half-cone in the vector space of matrices, many usual operations (like the mean) are stable in this space. However, negative eigenvalues appear as soon as one estimates the tensors from original data or when one uses standard numerical schemes for smoothing these data. We show that the choice of a convenient Riemannian metric allows to generalize consistently to tensor fields many important geometric data processing algorithms such as interpolation, filtering, diffusion and restoration of missing data. The methodology is exemplified on two important applications: the joint estimation and regularization of Diffusion Tensor MR Images (DTI), and the modeling of the variability of the brain from a dataset of precisely delineated anatomical structures (sulcal lines) in the cerebral cortex. In this context, we obtain a dense 3D variability map which proves to be in accordance with previously published results on smaller samples subjects. We also propose statistical tests which demonstrate that our model is globally able to recover the missing information and innovative methods to analyze the asymmetry of brain variability.
The talk will also address recent developments, including new Log-Euclidean metrics on tensors, that give a vector space structure to this manifold and a very efficient computational framework; Riemannian elasticity, a statistical framework on deformations fields for estimating the anatomical variability and regularizing accordingly in dense non-linear registration algorithms, and new clinical insights in scoliosis thanks to the statistical analysis of the anatomic variability of the spine.
References: (available at http://www-sop.inria.fr/epidaure/BIBLIO/Author/PENNEC-X.html)
* X. Pennec. Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements. To appear in J. of Math Imaging and Vision, 2006.
* X. Pennec, P. Fillard, and N. Ayache. A Riemannian Framework for Tensor Computing. International Journal of Computer Vision, 66(1):41-66, January 2006.

Brief Biography

Xavier Pennec holds an Engineering degree from the French Ecole Polytechnique in 1992. He also holds a PhD degree in Computer Science from the same institution in 1996, prepared at INRIA Sophia-Antipolis. After a post-doctoral fellowship at MIT (Mass. USA) in 1997, he was awarded in 1998 a research scientist position at the French National Institute for Research in Computer Science and Control (INRIA). His main research axes are about statistics on geometric data, in particular for medical image analysis, and biomedical image registration. Over the last years, these two fields have gradually converged toward computational anatomy, which aims at statistically describing the normal and abnormal shape of organs across populations. X. Pennec co-authored more than 20 international peer-reviewed journal papers and more than 50 international conference articles.