Geometric Statistics for Computational Anatomy

Abstract: At the interface of geometry, statistics, image analysis and medicine, computational anatomy aims at analysing and modelling the biological variability of the organs shapes and their dynamics at the population level. The goal is to model the mean anatomy, its normal variation, its motion / evolution and to discover morphological differences between normal and pathological groups. However, shapes are usually described by equivalence classes of sets of points, curves, surfaces or images under the action of a transformation group, or directly by the diffeomorphic deformation of a template in diffeomorphometry. This implies that they live in non-linear spaces, while statistics where essentially developed in a Euclidean framework. For instance, adding or subtracting curves or surfaces does not really make sense. Thus, there is a need for redefining a consistent statistical framework for objects living in manifolds and Lie groups, a field which is now called geometric statistics. The objective of this talk is to give an overview of the Riemannian computational tools and of simple statistics in these spaces. The talk is motivated and illustrated by applications in medical image analysis, such as the regression of simple and efficient models of the atrophy of the brain in Alzheimer’s disease and the groupwise analysis of the motion of the heart in sequences of images using the parallel transport of surface and image deformations.

machine learningMathematics

Audience: researchers in the discipline

Workshop on Geometry and Machine Learning