Description

The geometry in the title is not the geometry of lines and angles but the geometry of
topology, shape and knots. For example, galaxies are not distributed randomly in the
universe, but they tend to form clusters, or sometimes strings, or even sheets of high
galaxy density. How can this be handled statistically? The Euler characteristic (EC) of the
set of high density regions has been used to measure the topology of such shapes; it
counts the number of connected components of the set, minus the number of `holes’, plus
the number of `hollows'. Despite its complex definition, the exact expectation of the EC
can be found for some simple models, so that observed EC can be compared with
expected EC to check the model. A similar problem arises in functional magnetic
resonance imaging (fMRI), where the EC is used to detect local increases in brain activity
due to an external stimulus.
We are also interested the analysis of brain shape: does brain shape change with disease,
age or gender? Three types of data are now available: 3D binary masks, 2D triangulated
surfaces, and trivariate 3D vector displacement data from the nonlinear deformations
required to align the structure with an atlas standard. Again the Euler characteristic of the
excursion set of a random field is used to test for localised 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 nonisotropic, 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.
Finally we shall look in some detail at the statistical analysis of fMRI data. Our proposed
method seeks a compromise between validity, generality, simplicity and execution speed.
The method is based on linear models with local AR(p) errors fitted via the YuleWalker
equations with a simple bias correction that is similar to the first step in the Fisher scoring
algorithm for finding ReML estimates. The resulting effects are then combined across
runs in the same session, across sessions in the same subject, and across subjects within a
population by a simple mixed effects model. The model is fitted by ReML using the EM
algorithm after reparameterization to reduce bias, at the expense of negative variance
components. The residual degrees of freedom are boosted using a form of pooling by
spatial smoothing. Activation is detected using Bonferroni, False Discovery Rate, and
nonisotropic random field methods for local maxima and spatial extent. We conclude
with some suggestions for the optimal design of fMRI experiments.
