Detecting Changes in Brain Shape, Scale and Connectivity via the Geometry of Random Fields

Three types of data are now available to test for changes in brain shape: 3D binary masks, 2D triangulated surfaces, and trivariate 3D vector displacement data from the non-linear deformations required to align the structure with an atlas standard. We use the Euler characteristic of the excursion set of a random field as a tool to test for localized shape changes. We extend these ideas to scale space, where the scale of the smoothing kernel is added as an extra dimension to the random field. Extending this further still, we look at fields of correlations between all pairs of voxels, which can be used to assess brain connectivity. Shape data is highly non-isotropic, that is, the effective smoothness is not constant across the image, so the usual random field theory does not apply. We propose a solution that warps the data to isotropy using local multidimensional scaling. We then show that the subsequent corrections to the random field theory can be done without actually doing the warping - a result guaranteed in part by the famous Nash Embedding Theorem. This has recently been formalized by Jonathan Taylor who has extended Robert Adler's random field theory to arbitrary manifolds.