Abstract:

Current Issues in Statistical Analysis on Manifolds for Computational  
Anatomy

Xavier Pennec (mailto:Xavier.Pennec@sophia.inria.fr)
http://www.inria.fr/sophia/members/Xavier.Pennec/


Computational anatomy is an emerging discipline that aim at analysing  
and modeling the biological variability of the human anatomy. The  
method is to identify anatomically representative geometric features  
(points, tensors, curves, surfaces, volume transformations), and to  
describe and compare their statistical distribution in different  
populations. As these geometric features most often belong to  
manifolds that have no canonical Euclidean structure, we have to rely  
on more elaborated algorithmical basis.

I will first present the Riemannian structure, which proves to be  
powerfull enough to support a consistent framework for simple  
statistics on manifolds and can be extend to a complete computing  
framework on manifold-valued images. For instance, the choice of a  
convenient Riemannian metric on positive define symmetric matrices  
(tensors) 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  
is particularly well suited to the statistical estimation of Diffusion  
Tensor Images, and can also be used for modeling the brain variability  
from sulcal lines drawn at the surface of the cerebral cortex.

However, the goal is not just to model the variability of different  
types of anatomical features independently, but more importantly to  
model the variability of the underlying anatomy using all these  
observation means. Moreover, one would like to handle properly complex  
features such as curves and surfaces, which raises the problem of  
infinite dimensional manifolds. We will present here the approach  
based on currents developed by Stanley Durrleman during his PhD. This  
generative model combines a random diffeomorphic deformation model a  
al Grenander & Miller, that encodes the geometric variability of the  
anatomical template, with a random residual shape variability model (a  
la Kendall) on curves, sets of curves and surfaces encodes using  
currents.

In infinite dimensions, the question of the metric choice is crucial.  
In deformation-based morphometry, for instance, one may consider many  
different models of deformations, with potentially different results.  
We will shortly review some of them (static velocity fields,  
Riemannian elasticity) and, time permiting, finish with extensions of  
the previous methodologies to longitudinal evolution estimations in  
populations, which is currently becoming one of the very active topic.  
Through this guided tour of some of the current methods in  
computational anatomy, I will try to sketch the most important  
theoretical and practical challenges which could be the basis for  
future developments in information geometry.