Anisotropic Distributions on Manifolds: Template Estimation and Most Probable Paths

Construction of population atlases is a key issue in medical image analysis, and particularly in brain mapping. Large sets of images are mapped into a common coordinate system to study intra-population variability and inter-population differences, to provide voxel-wise mapping of functional sites, and help tissue and object segmentation via registration of anatomical labels. Common techniques often include the choice of a template image, which inherently introduces a bias. This paper describes a new method for unbiased construction of atlases in the large deformation diffeomorphic setting. A child neuroimaging autism study serves as a driving application. There is lack of normative data that explains average brain shape and variability at this early stage of development. We present work in progress toward constructing an unbiased MRI atlas of 2 years of children and the building of a probabilistic atlas of anatomical structures, here the caudate nucleus. Further, we demonstrate the segmentation of new subjects via atlas mapping. Validation of the methodology is performed by comparing the deformed probabilistic atlas with existing manual segmentations.

This paper describes the generation of large deformation diffeomorphisms phi:Omega=[0,1]3 Omega for landmark matching generated as solutions to the transport equation dphi(x,t)/dt=nu(phi(x,t),t),epsilon[0,1] and phi(x,0)=x, with the image map defined as phi(.,1) and therefore controlled via the velocity field nu(.,t),epsilon[0,1]. Imagery are assumed characterized via sets of landmarks {xn, yn, n=1, 2, ..., N}. The optimal diffeomorphic match is constructed to minimize a running smoothness cost parallelLnu parallel2 associated with a linear differential operator L on the velocity field generating the diffeomorphism while simultaneously minimizing the matching end point condition of the landmarks. Both inexact and exact landmark matching is studied here. Given noisy landmarks xn matched to yn measured with error covariances Sigman, then the matching problem is solved generating the optimal diffeomorphism phi;(x,1)=integral0(1)nu(phi(x,t),t)dt+x where nu(.)=argmin(nu.)integral1(0) integralOmega parallelLnu(x,t) parallel2dxdt +Sigman=1N[yn-phi(xn,1)] TSigman(-1)[yn-phi(xn,1)]. Conditions for the existence of solutions in the space of diffeomorphisms are established, with a gradient algorithm provided for generating the optimal flow solving the minimum problem. Results on matching two-dimensional (2-D) and three-dimensional (3-D) imagery are presented in the macaque monkey.