Random Spatial Structure of Geometric Deformations

The main goal of computational anatomy is to compare shapes of organs (e.g. brain, heart and spine) observed from computed tomography (CT) and magnetic resonance (MR) imaging. Statistical analysis of shape differences can be useful to understand disease-related changes of anatomical structures. The key idea is to estimate transformations between a template and patient anatomies. These transformations encode the structural differences in a population of patients. There is a wide range of groups of transformations that have been studied, ranging from rigid rotations to infinite dimensional groups of diffeomorphisms. Elements in these groups live on non-Euclidean constraint spaces such as manifolds. In this seminar, we present a selection of statistical challenges within the field of computational anatomy. We then outline our Bayesian approach to estimate spatial regions of interest that are common to a group of patients. Here, in contrast to traditional anatomy textbooks, the constituent parts of the anatomy are estimated from data. We propose a probabilistic model of the geometrical variability and describe individual patients as noisy deformations of a random spatial structure (modeled as regions) from a common template. The random regions are generated from a distance dependent Chinese Restaurant Process. We employ the Gibbs sampler to infer regions from a set of noisy deformation fields. Each step of the sampler involves model selection (Bayes factor) to decide about fusing regions. We show preliminary results on a dataset of spine CT images of patients that suffer from lower back and abdominal pain. This is joint work with Susan Holmes and Xavier Pennec.