CSAIL Event Calendar: Previous Series

Statistical Computing on Manifold Examplified with Tensors in DTI and Computational Anatomy

Speaker: Xavier Pennec , INRIA, Sophia Antipolis, France
Date: September 12 2008
Time: 2:00PM to 3:00PM
Location: 32-D507
Host: Polina Golland, CSAIL

In standard medical image analysis, many of the features extracted
from images or from their processing are geometric by nature and noisy
(e.g. transformations in registration, surfaces in
segmentations). Likewise, the usual method of computational anatomy,
an emerging discipline that aim at analysing and modeling the
biological variability of the human anatomy, is to identify
anatomically representative geometric features (points, tensors,
curves, surfaces, volume transformations), and to describe and compare
their statistical distribution in different populations.
Unfortunately, geometric features most often belong to manifolds that
are not vector spaces, which prevents the use of standard statistical
tools.

Based on a Riemannian manifold structure, we will detail how one can
develop a consistent framework for statistical computing on manifolds,
starting with the notions of mean value and covariance matrix of a
random element, normal law, Mahalanobis distance and test. Then, we
will extend the Riemannian computing framework to PDEs for smoothing
and interpolation of fields of geometric elements with the example of
positive define symmetric matrices (tensors). 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. This framework will be exemplified with the statistical
estimation of DTI images from DWI under a Rician noise assumption, and
with the modeling of the brain variability from a dataset of lines on
the cerebral cortex.

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