Statistical models of sets of curves and surfaces based on currents

Computing, visualizing and interpreting statistics on shapes like curves or surfaces is a real challenge with many applications ranging from medical image analysis to computer graphics. Modeling such geometrical primitives with currents avoids to base the comparison between primitives either on a selection of geometrical measures (like length, area or curvature) or on the assumption of point-correspondence. This framework has been used relevantly to register brain surfaces or to measure geometrical invariants. However, while the state-of-the-art methods efficiently perform pairwise registrations, new numerical schemes are required to process groupwise statistics due to an increasing complexity when the size of the database is growing. In this paper, we propose a Matching Pursuit Algorithm for currents, which allows us to approximate, at any desired accuracy, the mean and modes of a population of geometrical primitives modeled as currents. This leads to a sparse representation of the currents, which offers a way to visualize, and hence to interpret, such statistics. More importantly, this tool allows us to build atlases from a population of currents, based on a rigorous statistical model. In this model, data are seen as deformations of an unknown template perturbed by random currents. A Maximum A Posteriori approach is used to estimate consistently the template, the deformations of this template to each data and the residual perturbations. Statistics on both the deformations and the residual currents provide a complete description of the geometrical variability of the structures. Eventually, this framework is generic and can be applied to a large range of anatomical data. We show the relevance of our approach by describing the variability of population of sulcal lines, surfaces of internal structures of the brain and white matter fiber bundles. Complementary experiments on simulated data show the potential of the method to give anatomical characterization of pathologies in the context of supervised learning.