Template estimation form unlabeled point set data and surfaces for Computational Anatomy

A central notion in Computational Anatomy is the generation of registration maps,mapping a large set of anatomical data to a common coordinate system to study intra-population variability and inter-population differences. In previous work [1, 2] methods for estimating the common coordinate system or the template given a collection imaging data were presented based on the notion of Fréchet mean estimation using a metric on the space of diffeomorphisms. In this paper we extend the methodology to the estimation of a template given a collection of unlabeled point sets and surfaces. Using a representation of points and surfaces as currents a Reproducing Kernel Hilbert Space (RKHS) norm is induced on the space of Borel measures. Using this norm and a metric on the space of diffeomorphisms the template estimation problem is possed as a minimum mean squared error estimation problem. An efficient alternating conjugate gradient decent algorithm is derived and results exemplifying the methodology are presented.