We introduce two new types of random field. The cross correlation
field $R(\mathbf{s}, \mathbf{t})$ is the usual sample correlation coefficient
for a set of pairs of Gaussian random fields, one sampled at point $s \epsilon
\Re^M$, the other sampled at point $\mathbf{t} \epsilon \Re^N$. The homologous
correlation field is defined as $R(\mathbf{t}) = R(\mathbf{t}, \mathbf{t})$,
that is, the "diagonal" of the cross correlation field restricted to
the same location $\mathbf{s} = \mathbf{t}$. Although the correlation
coefficient can be transformed pointwise to a t-statistic, neither of
the two correlation fields defined above can be transformed to a
t-field, defined as a standard Gaussian field divided by the root mean
square of i.i.d. standard Gaussian fields. For this reason, new results are
derived for the geometry of the excursion set of these correlation fields that
extend those of Adler. The results are used to detect functional connectivity
(regions of high correlation) in three-dimensional positron emission tomography
(PET) images of human brain activity.