Optimal data-driven sparse parameterization of diffeomorphisms for population analysis

In this paper, we propose a novel approach for intensity based atlas construction from a population of anatomical images, that estimates not only a template representative image but also a common optimal parameterization of the anatomical variations evident in the population. First, we introduce a discrete parameterization of large diffeomorphic deformations based on a finite set of control points, so that deformations are characterized by a low dimensional geometric descriptor. Second, we optimally estimate the position of the control points in the template image domain. As a consequence, control points move to where they are needed most to capture the geometric variability evident in the population. Third, the optimal number of control points is estimated by using a log - L¹ sparsity penalty. The estimation of the template image, the template-to-subject mappings and their optimal parameterization is done via a single gradient descent optimization, and at the same computational cost as independent template-to-subject registrations. We present results that show that the anatomical variability of the population can be encoded efficiently with these compact and adapted geometric descriptors.