  The permutation test is an often used test procedure in brain imaging.
Unfortunately, generating every possible permutation for large-scale brain
image datasets such as HCP and ADNI with hundreds images is not practical. Many
previous attempts at speeding up the permutation test rely on various
approximation strategies such as estimating the tail distribution with known
parametric distributions. In this study, we show how to rapidly accelerate the
permutation test without any type of approximate strategies by exploiting the
underlying algebraic structure of the permutation group. The method is applied
to large number of MRIs in two applications: (1) localizing the male and female
differences and (2) localizing the regions of high genetic heritability in the
sulcal and gyral pattern of the human cortical brain.
