Detecting Shape Changes via Non-Isotropic Random Fields

Three types of data are now available to test for changes in brain shape: 3D binary data for the indicator function or mask of the structure; 2D displacement data from the surface of the 3D structure; 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 structural changes using local maxima and size of clusters in the excursion set. The 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 – it is only sufficient to know that such a warp exists – a fact that is guaranteed in part by Nash’s Embedding Theorem. We shall apply thee methods to a set of 151 brain images from the Human Brain Mapping data base.