We are all familiar with the correlation coefficient between two sets of numbers. Now suppose we replace the numbers by vector-valued images in any number of dimensions. The correlation random field is the 'image' of correlations at all possible pairs of points in the two images. We use random field theory to set a threshold on the correlations so that those above the threshold are statistically significant, corrected for searching over all pairs of points. We apply this idea to resting state networks of fMRI images of brain activity, and networks of connectivity in cortical thickness.

We present a unified computational approach to tensor-based morphometry in detecting the brain surface shape difference between two clinical groups based on magnetic resonance images. Our approach is novel in a sense that we combined surface modeling, surface data smoothing and statistical analysis in a coherent unified mathematical framework. The cerebral cortex has the topology of a 2D highly convoluted sheet. Between two different clinical groups, the local surface area and curvature of the cortex may differ. It is highly likely that such surface shape differences are not uniform over the whole cortex. By computing how such surface metrics differ, the regions of the most rapid structural differences can be localized. To increase the signal to noise ratio, diffusion smoothing based on the explicit estimation of Laplace-Beltrami operator has been developed and applied to the surface metrics. As an illustration, we demonstrate how this new tensor-based surface morphometry can be applied in localizing the cortical regions of the gray matter tissue growth and loss in the brain images longitudinally collected in the group of children.