The value of diffusion imaging lies not only in its promise to recover
patterns of connectivity in the human brain, but also to quantify and
study the structure of anisotropy through-out the image. At its
simplest form, this involves detecting changes in fractional
anisotropy (FA) correlated with neurological conditions. More
generally, the shape of anisotropic regions can be studied by
extracting a skeleton of the white matter as a set of sheets or tubes.
Previous work on extracting "anisotropy creases" (ridges and valleys)
has recently been extended into a linear scale-space, using properties
of Lindeberg's discrete Gaussian for interpolating in scale, and using
particle systems for computing a uniform sampling of the anisotropic
features through space and scale.
It is a non-trivial statement about
the biological structure of anisotropy that its ridge surfaces and
ridge lines tend to be aligned with the eigenvectors of the underlying
diffusion tensor; there is no mathematical reason why this must be so.
Current work quantifies the orientation alignment between the Hessian
of diffusion anisotropy and the tensor itself, as parameterized by a
particular quotient space of the quaternions. A consequence of this
may be an automatic way of detecting certain errors in the coordinate
systems used to describe diffusion imaging experiments.