  High angular resolution diffusion imaging data is the observed characteristic
function for the local diffusion of water molecules in tissue. This data is
used to infer structural information in brain imaging. Nonparametric scalar
measures are proposed to summarize such data, and to locally characterize
spatial features of the diffusion probability density function (PDF), relying
on the geometry of the characteristic function. Summary statistics are defined
so that their distributions are, to first-order, both independent of nuisance
parameters and also analytically tractable. The dominant direction of the
diffusion at a spatial location (voxel) is determined, and a new set of axes
are introduced in Fourier space. Variation quantified in these axes determines
the local spatial properties of the diffusion density. Nonparametric hypothesis
tests for determining whether the diffusion is unimodal, isotropic or
multi-modal are proposed. More subtle characteristics of white-matter
microstructure, such as the degree of anisotropy of the PDF and symmetry
compared with a variety of asymmetric PDF alternatives, may be ascertained
directly in the Fourier domain without parametric assumptions on the form of
the diffusion PDF. We simulate a set of diffusion processes and characterize
their local properties using the newly introduced summaries. We show how
complex white-matter structures across multiple voxels exhibit clear
ellipsoidal and asymmetric structure in simulation, and assess the performance
of the statistics in clinically-acquired magnetic resonance imaging data.
