White Matter Fiber Analysis

Shape, scale, orientation and position, the physical features associated with white matter fibers, can, either individually or in combination, be used to define feature spaces designed for specific end-applications. Such a treatment is useful since the quantitative analysis of white matter fibers has diverse applications, each with a different focus and objective.

In recent work, we describe a Riemannian framework in which various combinations of these features are considered. (This was presented at the ISBI 2010 conference. The slides are here, a version of the paper here.)

The framework also provides tools for computing statistical summaries of curves which enables us to perform a full statistical analysis. In the context of DTI fibers, a mean and variance that describes the essential characteristics of the fiber bundle can be used to represent a set of fibers. We can then proceed to tasks of statistical inference such as parameter estimation and hypothesis testing.

I am currently using the tools and metrics defined within this mathematical framework to show how morphological changes due to disease progression can be studied. Shape distances in tandem with distances defined within other manifolds like the shape+orientation manifold give us very encouraging results.

Differential geometry in 10 slides

Partha Niyogi's very lucid talk entitled Geometric Methods and Manifold Learning includes a brief and very basic introduction to differential geometry(starts at t=40:49) which I found helpful.

This was part of the Machine Learning Workshop I attended at the University of Chicago last June (MLSS'09). There were several other talks and tutorials of note. I especially enjoyed Emmanuel Candes' talk on sparse signal recovery. The talks are available at the videolectures website.